Magnus Blomkvist
email: mblomkvist@audencia.com

Timo Korkeamäki
email: timo.korkeamaki@aalto.fi

Tuomas Takalo
email: tuomas.takalo@bof.fi

Wolfgang Bessler

Gonul Colak

Espen Eckbo

Alexandre Garel

Abhinav Goyal

Daniel Hauser

Peter Nyberg

Sami Torstila

Raman Uppal

Mark Walker

Per Östberg Kam-Ming Wang

Learning and staged equity financing*

Keywords: G14, G24, G32 Keywords: IPOs, security issuance, sequential financing G14, G24, G32 Keywords: IPOs, security issuance, sequential financing Observations Observations

We propose a rationale for why firms often return to the equity market shortly after their initial public offering (IPO). We argue that hard to value firms conduct smaller IPOs, and that they return to the equity market conditional on a positive valuation signal. This is driven by two-way learning, as market information complements both corporate disclosure and internal information available to management. In contrast to prior studies, we find that information asymmetry is not a necessary condition for staged financing. Our arguments receive support in a sample of 3,625 U.S. IPOs between 1980-2018.

Introduction

A large number of firms return to equity market soon after their initial public offering (IPO). [START_REF] Hertzel | Public Market Staging: The Timing of Capital Infusions in Newly Public Firms[END_REF] report that such behavior is often premeditated, as witnessed by disclosures in the IPO prospectuses. While many of the existing explanations for staged financing rest upon information asymmetry, we offer a simpler rationale for it. Our analysis suggests that using the IPO and the subsequent trading information as learning channels is sufficient to explain the observed sequential equity financing behavior of IPO firms, even in the absence of information asymmetry. The owner-manager deliberately undertakes a relatively small IPO, in order to gain information that will guide her subsequent capital raising decisions. This is consistent with [START_REF] Lowry | Is the IPO Pricing Process Efficient?[END_REF], and [START_REF] Loughran | Why don't Issuers Get Upset About Leaving Money on The Table in IPOs?[END_REF], who find that the IPO process by itself fails to fully incorporate all public information. 1 Firms thus have an incentive to conduct their IPO in stages, especially when they operate in an environment with high uncertainty.

Our work is inspired by the literature on the feedback effects of stock prices. 2 This literature indicates that stock market trading provides valuable information that guides firm managers in taking actions that affect the value of their firms. New to the literature, we apply this argument in a setting where the firm raises equity through an IPO. As we note above, we posit that staging of equity offerings allows the firm's owner-manager to use the initial stock market information to learn about the valuation of the firm and then calibrate the size and the timing of a subsequent equity offering based on this information.

While our main contribution is an empirical one, we develop a parsimonious model that illustrates the idea behind our work. Inspired by [START_REF] Holmström | Managerial Incentive Problems -A Dynamic Perspective[END_REF], [START_REF] Aghion | Innovation and Institutional Ownership[END_REF], and [START_REF] Dow | Incentives for Information Production in Markets Where Prices Affect Real Investment[END_REF], we build upon two-sided learning regarding market value of the firm. First, depending on the initial valuation signal and uncertainty, the owner chooses both whether to conduct an IPO and the size of the IPO. If valuation uncertainty is low and the initial valuation is high (low), the manager conducts a large IPO (no IPO). In situations where the manager faces a high degree of uncertainty, she initially conducts a small IPO.

1 Also, several studies report substantial inefficiencies in IPO pricing. See e.g. [START_REF] Aussenegg | IPO pricing with bookbuilding and a when-issued market[END_REF], [START_REF] Derrien | IPO pricing in "hot" market conditions: Who leaves money on the table?[END_REF], Ljunqvist and Wilhelm (2003), Purnanandam and Swaminathan (2005). 2 See, e.g. [START_REF] Edmans | Feedback Effects, Asymmetric Trading, and the Limits to Arbitrage[END_REF], [START_REF] Dow | Incentives for Information Production in Markets Where Prices Affect Real Investment[END_REF] for theoretical arguments, and [START_REF] Edmans | The Source of Information in Prices and Investmentprice Sensitivity[END_REF] for empirical evidence. Also, [START_REF] Bond | The Real Effects of Financial Markets[END_REF] provide an early survey of the literature.

If the post-IPO return is positive, the firm returns to the equity market to conduct an SEO to raise additional equity capital. In the model, both the IPO process and the subsequent trading activity transmit information, which motivates the firm's follow-on SEO issue. Our model is not unique in the sense that several IPO models build on two sided learning, where the managers learn about valuation of their firms from the market (see e.g., [START_REF] Benveniste | How Investment Bankers Determine the Offer Price and Allocation[END_REF][START_REF] Dow | Stock Market Efficiency and Economic Efficiency: Is There a Connection?[END_REF][START_REF] Chemmanur | A Theory of the Going-Public Decision[END_REF][START_REF] Benveniste | Information Externalities and the Role of Underwriters in Primary Equity Markets[END_REF]Alti, 2005;[START_REF] Hsieh | A Theory of Merger Driven IPOs[END_REF]. Our symmetric-but-incomplete information assumption is not only a simple way to explain follow-on SEOs but it is also plausible in the early stage trading where uncertainty about the firm value is high.

Many of the existing theories of SEOs, such as [START_REF] Allen | Signaling by Underpricing in the IPO Market[END_REF] and [START_REF] Welch | Seasoned Equity Offerings, Imitation Costs, and the Underpricing of Initial Public Offerings[END_REF], are based on the traditional perception (see [START_REF] Myers | Corporate Financing and Investment Decisions when Firms Have Information that the Investors Do not Have[END_REF]) that a firm's owner-manager is better informed about the firm's market value than the market. While the owner-manager may well have private information about some internal aspects of the firm, it is equally conceivable that she faces uncertainty about the market valuation prior to an IPO. For information asymmetry to drive the sequential financing behavior, the owner-manager would essentially need to be better informed than the market about the value of her firm.

Much of the empirical literature on staged financing also relies on agency costs as a driver. Among venture capital firms, [START_REF] Gompers | Optimal Investment, Monitoring, and the Staging of Venture Capital[END_REF] and [START_REF] Sahlman | The Structure and Governance of Venture-Capital Organizations[END_REF] report that concerns for agency issues and subsequent credit rationing [START_REF] Stiglitz | Credit Rationing in Markets with Imperfect Competition[END_REF] tend to affect the design of financing rounds. Staged financing can also be carried out through security issues such as convertible bonds [START_REF] Mayers | Why Firms Issue Convertible Bonds: The Matching of Financial and Real Investment Options[END_REF] or unit offerings [START_REF] Schultz | Unit Public Offerings: A Form of Staged Financing[END_REF]. With design of hybrid securities, incentives can be set for management to both invest first-stage proceeds efficiently, and provide more accurate information (Korkeamäki and Moore, 2004;[START_REF] Cornelli | Stage Financing and the Role of Convertible Securities[END_REF][START_REF] Schultz | Unit Public Offerings: A Form of Staged Financing[END_REF].

As we hypothesize that uncertainty regarding firm valuation has an important effect on the firm's likelihood to rely on sequential financing, we develop metrics for hard to value (HTV) firms. We use the principal component analysis (PCA), which allows us to capture different aspects of challenges in firm valuation, as indicated by prior studies. Our main PCA metric is based on the three variables suggested by [START_REF] Gompers | Optimal Investment, Monitoring, and the Staging of Venture Capital[END_REF], namely R&D expenses, firm age, and an indicator for high tech industries. For robustness, we define a second PCA metric, that uses two additional variables, price revisions during the underwriting process, and an indicator for negative earnings. All of our PCA inputs are widely used as proxies for difficulty to value (see e.g. [START_REF] Lowry | The Variability of IPO Initial Returns[END_REF][START_REF] Hertzel | Public Market Staging: The Timing of Capital Infusions in Newly Public Firms[END_REF][START_REF] Colak | Political Uncertainty and IPO Activity: Evidence from U.S. Gubernatorial Elections[END_REF]. With the exception of the price revision indicator, we observe each of the other four variables at the end of the fiscal year prior to the IPO. In both PCA settings, we obtain the first principal component as our measure of hard to value.

It is notable that studies on financial slack relate firms' need of additional slack to some of the same variables that are behind our PCA-based metric on hard-to-value. For instance, [START_REF] Leary | The Pecking Order, Debt Capacity, and Information Asymmetry[END_REF] suggest that older firms have less demand for slack, and firms with greater investment opportunities (generated by R&D) require more slack [START_REF] Denis | Persistent Negative Cash Flows, Staged Financing, and the Stockpiling of Cash Balances[END_REF]. In our empirical analysis, these slackrelated predictions generate a bias against findings, as additional need of slack should motivate larger IPO injections of capital, so that young and R&D intensive firms would prefer large IPOs. In contrast, our expectation is that as proxies of difficulty to value, the metrics above are connected to smaller IPO Size and subsequent SEOs.

We test the implications of our model in a sample of 3,625 US firms that complete an IPO during 1.1.1980-31.12.2018. 3 Consistent with our expectations, we find a strong negative connection between difficulty to value and the IPO size. We further report a positive relation between IPO size and the likelihood of a follow-on SEO within two years of the IPO. Both of these findings align with the idea that firms with difficult to value projects fulfil their capital needs in stages. Furthermore, a hazard model analysis indicates that difficulty to value shortens the time to SEO for IPO firms. When we observe the connections between difficulty to value, IPO Size, and stock returns of IPO firms, we find that in support of our model's implications, firms with smaller IPOs and good outcomes following the IPO are more likely to follow with an SEO within two years. Our results are consistent with several studies that argue that managers learn from stock prices (see e.g., [START_REF] Bond | The Real Effects of Financial Markets[END_REF][START_REF] Edmans | Feedback Effects, Asymmetric Trading, and the Limits to Arbitrage[END_REF][START_REF] Edmans | The Source of Information in Prices and Investmentprice Sensitivity[END_REF]. Also, survey evidence by [START_REF] Brau | Initial Public Offerings: An Analysis of Theory and Practice[END_REF] indicates that one of the main reasons for a firm to go public is to resolve uncertainty about its valuation.

We subject our findings to a number of robustness tests. In a part of our empirical analysis, we relax the assumption of fully asymmetric information structure in the IPO, and observe the participation of insiders in the IPO and follow-on offerings. Our findings on insider behavior during the IPO support the notion that asymmetric information does not drive our results. Market timing plays a role in firms' security issuance decisions [START_REF] Baker | Market Timing and Capital Structure[END_REF][START_REF] Taggart | A Model of Corporate Financing Decisions[END_REF]. Market timing and clustering of IPOs into periods of "hot" markets is therefore another potential motive for raising equity in stages, and we consider it as an alternative explanation for follow-on SEOs in our empirical tests. Concern for dilution during cold markets may motivate the firm to postpone some of its current projects until a subsequent financing round [START_REF] Myers | Corporate Financing and Investment Decisions when Firms Have Information that the Investors Do not Have[END_REF][START_REF] Strebulaev | Optimal Issuance under Asymmetric Information and Accumulation of Cash Flows[END_REF]. While we include controls for IPO market cycles in all our regression tests, we further study whether market timing has an effect on the dynamics of the sequential equity financing pattern that we document, and we find no such pattern. Furthermore, our empirical findings could arise from selection bias, as only a fraction of HTV firms may enter the IPO market and, as predicted by our model, poor quality easy-to-value firms may not get listed. To alleviate this concern, we use propensity score matching and study separately a sample of HTV and ETV firms with matched quality characteristics. HTV firms issue smaller IPOs and are more likely to return to the equity market in that setting as well. Our results are also robust to exclusion of VC-backed issues from the sample. This suggests that our findings are not driven by a bias arising from only a fraction of VC-backed firms going public.

In auxiliary tests, we explore how the market feedback channel holds up in relation to signaling models [START_REF] Allen | Signaling by Underpricing in the IPO Market[END_REF]Welch, 1990) and negative cash flow explanations [START_REF] Denis | Persistent Negative Cash Flows, Staged Financing, and the Stockpiling of Cash Balances[END_REF][START_REF] Huang | Corporate Cash Shortfalls and Financing Decisions[END_REF]. According to the signaling models, high quality firms conduct smaller underpriced IPOs to subsequently issue an SEO. Our results remain intact after inclusion of underpricing in the IPO size regressions, which indicates that our results are not driven by signaling efforts. Also, recent evidence by [START_REF] Denis | Persistent Negative Cash Flows, Staged Financing, and the Stockpiling of Cash Balances[END_REF] suggests that staged equity financing is motivated by cash needs stemming from long-term negative cash flows. We are taking this potential explanation into account, as our HTV measure includes a negative earnings indicator. Even though cash flows are negatively related to the SEO likelihood, our results remain intact after the inclusion of a cash flow variable.

Besides the information produced by their own market entry, IPO firms can also use the alternative learning channel of cross learning from industry peers [START_REF] Aghamolla | IPO Peer Effects[END_REF][START_REF] Foucault | Learning From Peers' Stock Prices and Corporate Investment[END_REF][START_REF] Kaustia | Social Learning and Corporate Peer Effects[END_REF]. Informative peers may thus aid high uncertainty peers to enter the market.

This idea receives support in our tests, which suggests that the HTV firms with low peer informativeness and peer valuations stay out of the market. Our results may thus be biased downwards, as we do not observe firms that are the most difficult-to-value.

We are not the first ones to point out the market feedback channel as an explanation for staged equity financing as [START_REF] Jegadeesh | An Empirical Investigation of IPO Returns and Subsequent Equity Offerings[END_REF] suggest such channel in their early work. However, the channel has not received much research attention, as the literature has focused on the asymmetric information explanation. [START_REF] Jegadeesh | An Empirical Investigation of IPO Returns and Subsequent Equity Offerings[END_REF] interpret the positive relation between post-IPO stock returns and the SEO likelihood as staged financing that is driven by market feedback. However, their setting makes it very difficult to distinguish between pre-meditated staged financing and traditional market timing behavior. We explicitly test for how the level of valuation uncertainty combined with positive post-IPO return realizations increases the SEO likelihood, while we also exclude the market timing channel.

Our paper is empirically closest to [START_REF] Hertzel | Public Market Staging: The Timing of Capital Infusions in Newly Public Firms[END_REF], who observe financing injections within two years of the IPO. They motivate staging from the information asymmetry standpoint, and view it arising either from markets' reluctance to provide capital due to agency concerns, or from the firm's incentive to time their funding, based on inside information about future prospects. The results in [START_REF] Hertzel | Public Market Staging: The Timing of Capital Infusions in Newly Public Firms[END_REF] are inconclusive as to whether supply-or demand-driven explanations dominate in motivating sequential financing, but they note the important role of agency issues, as they constrain availability of sufficient equity capital at the time of the IPO. Also, [START_REF] Cole | Exchange Upgrades: Reducing Uncertainty through a Two-Stage IPO[END_REF] address the effects of information learned from the market around the time of the IPO. They observe firms that are listed on the over-the-counter market prior to their IPO. They find that post-IPO uncertainty, measured by both stock return volatility and textual content in corporate disclosures, is lower for firms that have a prior over-the-counter market listing. Furthermore, Derrien and Kecskes (2007) find in a UK setting that in the presence of valuation uncertainty, some firms conduct IPOs raising zero equity proceeds and subsequently return to raise funding from the market after a reduction in valuation uncertainty. Consistent with the predictions of our study, pre-IPO listing thus appears to have a two-sided effect, as it reduces uncertainty both for the issuing firm and for the investors. We contribute by analyzing the relevance of this learning channel even at the point when firms consider going public. A key implication of our study is that an IPO process that transmits information effectively is conductive for a subsequent equity offering. While Hertzel, et al. (2012) consider all financing injections subsequent to the IPO, our focus is on equity financing. This focus is motivated by equity valuation being more sensitive to information than debt valuation (see, e.g., [START_REF] Dang | Banks as Secret Keepers[END_REF][START_REF] Mclean | The Business Cycle, Investor Sentiment, and Costly External Finance[END_REF]. We thus exclude from our analysis any considerations where the initial IPO would be followed by increased leverage and more limited dilution, and concentrate on a two-sided learning effect under information symmetry, where the original desired level of equity financing (and dilution) is fixed. We further note that our exclusive focus on equity issuance may also bias our empirical tests against findings. Debt tends to be more accessible to firms that are easy to value, and furthermore, HTV firms with positive valuation signals after their IPOs should also enjoy an improved access to debt financing. Existence of debt would thus decrease the size of IPOs for easy-to-value firms, and make subsequent equity issues less appealing for HTV firms with good post IPO performance.

A model of staged equity financing

Basic framework

Consider a privately-held firm whose controlling owner ("the owner") is contemplating a public equity offering. There are two periods, t ∈ {1, 2}. At the beginning of period 1, the owner decides first whether or not to sell a stake of the firm via a stock market and, upon the IPO decision, the size of the IPO.

If the owner decides to make a small IPO, the owner will then at the beginning of period 2 decide whether or not to make a follow-on SEO. For simplicity, we normalize the total number of shares the owner is contemplating to offer to two, and denote the owner's equity offering decisions by at, with a1 ∈ {0,1,2} and a2 ∈ {0,1}. We denote by 𝛿 ∈ [0,1] the owner's discount factor between the periods, capturing the benefits of financial slackthe preference for having the cash in hand sooner rather than later.

Following, e.g., [START_REF] Holmström | Managerial Incentive Problems -A Dynamic Perspective[END_REF], [START_REF] Aghion | Innovation and Institutional Ownership[END_REF][START_REF] Dow | Incentives for Information Production in Markets Where Prices Affect Real Investment[END_REF], there is incomplete but symmetric information about the intrinsic value of the firm as a public company. The intrinsic value of the firm (per share) 𝜃 ∈ ℝ + can be either high θH with probability 𝑝 ∈ (0,1) or low θL, θH > θL, with probability 1-p. We may think of θ as representing talent of the firm's manager (to run a publicly listed company) as in [START_REF] Holmström | Managerial Incentive Problems -A Dynamic Perspective[END_REF] and [START_REF] Aghion | Innovation and Institutional Ownership[END_REF]. The parameter p reflects whether it is easy to value the firm or not: The firms with a very high or a very low p are easy to value whereas the firms with the intermediate values of p are difficult to value.

If the owner decides to go public, an IPO produces an imperfect signal S of the firm's intrinsic value θ during period 1. Here the term "IPO" can be interpreted loosely in the sense that includes both the preparation for the IPO and the stock market trading in the immediate aftermath. More specifically, S = h (respectively, S = l) with probability q ∈ [1/2,1] when θ = θH (θ = θL) i.e., the signal reveals correct information about the firm's intrinsic value with probability q and is misleading with probability 1-q. The parameter q captures the information quality of the IPO. If q = 1/2, the IPO produces no additional information about the firm's intrinsic value and if q = 1, and the IPO is perfectly revealing.

As in [START_REF] Zingales | Insider Ownership and the Decision to Go Public[END_REF], we assume that the firm produces a non-verifiable private benefit of 𝐵 ∈ ℝ + to the owner. This benefit is lost if the owner decides to sell her share. There are a number of reasons for owners to waive the possibility of selling their stock to the public market, such as loss of the private benefits of control and freedom to operate without a pressure from external shareholders [START_REF] Dyck | Private Benefits of Control: An International Comparison[END_REF]. 4 For simplicity we assume that the owner's private benefits are proportional to her control rights and hence 4 Other reasons for owners to stay private include the possibility of keeping business plans and financing sources confidential, and the desire to avoid laborious regulatory and reporting requirements of publicly listed companies. These opportunity costs of public listing are fully realized upon an IPO irrespective of its size. On the other hand, the private benefits of control might be lost only after the owner relinquishes a majority or all of her control rights. In this case, a small IPO would not realize these costs. We could allow for larger opportunity costs of public listing upon an IPO or only when the owner sells (the majority of) her company at the cost of complicating the notation. Yet one B is measured per share. Under this assumption the owner's private value of the firm's share remains unchanged irrespective of chosen a1. Finally, the transaction cost of a public offering is 𝐹 ∈ ℝ + irrespective of its size. We may think B captures all opportunity costs of a public offering that are proportional to the size of the offering and F captures fixed costs of the offering. The owner's objective is to maximize the firm's value conditional on the information available to investors. The timing of events is as follows: In period 1, the owner chooses the size of an IPO, a1 ∈ {0,1,2}.

In the cases where a1 = 0 (the owner decides to remain private) or a1 = 2 (the owner decides to issue a single, large IPO), the owner's decision sequence ends. Upon the IPO, the signal S is generated, and the market updates its beliefs about the firm value. In period 2, which is relevant only if a1 = 1, the owner, after observing the firm's post-IPO market value, chooses the size of a SEO, 𝑎 2 (𝑎 1 = 1, 𝑠) ≔ 𝑎 2 (𝑠) ∈ {0,1}.

Listing decisions

Prior to an IPO, the expected market valuation of a firm's stock is given by (1) E(𝑉) = 𝑝𝜃 𝐻 + (1 -𝑝)𝜃 𝐿 .

While we follow [START_REF] Zingales | Insider Ownership and the Decision to Go Public[END_REF] to assume that the firm's expected market valuation is independent of the owner's private benefits from her control rights, the desirability of a public offering depends on the relation between E(V) and B. For example, if B is very large compared to E(V), the owner will not conduct an IPO.

A special case of our model is that 𝐵 = E(V), i.e., the expected market valuation of the firm's stock exactly equals the owner's private valuation of her control rights.

After the IPO, the market updates its beliefs about the firm value using the signal realization S = s and Bayes' rule. Let Pr(θ|s) denote the probability that the firm's intrinsic value is 𝜃 ∈ {𝜃 𝐿 , 𝜃 𝐻 }, given a signal realization 𝑠 ∈ {𝑙, ℎ}, and let V denote the firm's market valuation. Then, the firm's post-IPO market opportunity cost of a public offering is the possibility to sell out to a private buyer. This opportunity cost can be considered to be proportional to the size of the public offering -see [START_REF] Zingales | Insider Ownership and the Decision to Go Public[END_REF].

valuations are given by 𝑉 ℎ ≔ 𝑉|ℎ = Pr(𝜃 𝐻 |ℎ)𝜃 𝐻 + Pr(𝜃 𝐿 |ℎ)𝜃 𝐿 , and 𝑉 𝑙 ≔ 𝑉|𝑙 = Pr(𝜃 𝐻 |𝑙)𝜃 𝐻 + Pr(𝜃 𝐿 |𝑙)𝜃 𝐿 , which can be rewritten by using Bayes' rule as

(2) 𝑉 ℎ = 𝑞𝑝𝜃 𝐻 + (1 -𝑞)(1 -𝑝)𝜃 𝐿 𝑞𝑝 + (1 -𝑞)(1 -𝑝) ,

and

(3)

𝑉 𝑙 = (1 -𝑞)𝑝𝜃 𝐻 + 𝑞(1 -𝑝)𝜃 𝐿 𝑞(1 -𝑝) + (1 -𝑞)𝑝 .
If the signal is informative (q > 1/2), Vh. > E(V) > VL.

To make the IPO decision meaningful, we impose the following assumption on the parameters:

Assumption 1. Vh > B + F > Vl.
Assumption 1 implies that if the post-IPO market valuation is high (low) the owner will (not) sell another stake at the market. Since Pr(θ|s) ∈ [0,1] for all 𝜃 and s, θH ≥ Vh and Vl ≥ θL (with the inequalities being strict unless q = 1). Hence, Assumption 1 also implies that θH > B+ F > θL. In words, if it were certain that the firm is of high (low) value, the owner would prefer (not) to list publicly. To make notation more compact, we introduce the definitions ΔH := θH -B and ΔL := B -θL reflecting the relation of the firm's intrinsic value to the owner's private valuation of her control rights. Here our assumptions imply ∆ 𝐻 ∈ (𝐹, ∞) and ∆ 𝐿 ∈ (-𝐹, 𝐵]. Finally, to establish analytic results compactly, we assume that F is "small" otherwise we would need to consider subcases where a large IPO or a staged equity offering is unprofitable.

We write the firm's expected net value for the owner conditional on the IPO decision as V(a1). Our aim is to characterize the circumstances when planning a staged equity offering is optimal for the owner,

V(1) ≥ max {V(2),V(0)}.
The expected value of a single, large IPO is given by ( 4)

𝑉(2) = 2[Pr(ℎ)𝑉 ℎ + Pr(𝑙)𝑉 𝑙 -𝐵] -𝐹,
in which the probabilities of high and low market valuations following the IPO are Pr(h) = qp+(1-q)(1p) and Pr(l) = q(1-p)+p(1-q), respectively. The term in the square brackets gives the owner's expected net payoff per share from going to public. Using the fact that Pr(h)Vl + Pr(l)Vl = E(V), equation ( 4) can be simplified to ( 5)

𝑉(2) = 2(E(𝑉) -𝐵) -𝐹.
Thus, V(2) ≥ 0 only if E(V) ≥ B + F/2. Under this condition, a large IPO is more profitable for the owner than keeping her firm private. Note that the owner requires a premium to conduct a large IPO: A necessary condition for V(2) ≥ 0 is that the expected market valuation of the firm's stock is strictly larger than the owner's private valuation of her control rights (E(V) > B). Using equation ( 1) we may rewrite the condition

E(V) ≥ B + F/2 as (6) 𝑝 ≥ 𝑝 ̃≔ 𝐹 + 2 ∆ 𝐿 2(∆ 𝐻 + ∆ 𝐿 )
in which 𝑝 ̃∈ (0,1).

The owner's expected payoff to a staged equity offering with two smaller rounds is ( 7)

𝑉(1) = Pr(ℎ) (𝑉 ℎ + 𝛿𝑉 2 (ℎ)) + Pr(𝑙) (𝑉 𝑙 + 𝛿𝑉 2 (𝑙)) -𝐵 -𝐹,
in which V2(s) := V2(a2(s),s) gives the owner's expected net payoff from a SEO. Since Assumption 1 implies that the owner issues an SEO only upon a high signal (a2(l) = 0 and a2(h) = 1), we have V2(l) = 0 and V2(h) = 𝑉 ℎ -𝐵 -𝐹. We can thus simplify equation ( 7) to ( 8) 𝑉(1) = 𝐸(𝑉) -𝐵 -𝐹 + Pr(ℎ) 𝛿(𝑉 ℎ -𝐵 -𝐹).

The first three terms in the right hand side of equation ( 8) capture the owner's expected net payoff from the IPO, and the last term captures the expected value of the option to the SEO in the case the firm's market valuation turns out to be high.

Planning a staged equity offering is more profitable for the owner than a large IPO when V(1) ≥ V(2). After some algebra using equations ( 1), ( 2), ( 5) and ( 8), we get that the condition

V(1) ≥ V(2) is equivalent to (9) 𝑝 ≤ 𝑝̅ ≔ ∆ 𝐿 -𝛿(1 -𝑞)(∆ 𝐿 + 𝐹) ∆ 𝐻 + ∆ 𝐿 -𝛿[(1 -𝑞)(∆ 𝐿 + 𝐹) + 𝑞(∆ 𝐻 -𝐹)]
.

Our assumptions imply that 𝑝̅ < 1. (We have 𝑝̅ = 1 only if δ = q = 1 and F = 0.)

Next, for the owner, planning a staged equity offering is preferable to staying private if V(1) ≥ 0.

After substituting equations ( 1) and (2) for equation ( 8) we get that V(1) ≥ 0 if

(10) 𝑝 ≥ 𝑝 ≔ (∆ 𝐿 + 𝐹)[1 + 𝛿(1 -𝑞)] ∆ 𝐻 + ∆ 𝐿 + 𝛿[(1 -𝑞)(∆ 𝐿 + 𝐹) + 𝑞(∆ 𝐻 -𝐹)]
.

Under Assumption 1, 𝑝 ∈ (0,1).

From inequalities ( 6), ( 9) and ( 10) we observe that if F is sufficiently small, 0 < 𝑝 < 𝑝 ̃< 𝑝̅ < 1.

Hence, for 𝑝 ∈ (𝑝, 𝑝̅ ), it is optimal for the owner to plan a staged equity offering. Such a staged offering allows the owner to experimentto learn about the market value of her firmand gives the option to follow with an SEO in case the market valuation turns out to be high. The fixed cost of public offerings (F) needs to be sufficiently small to make experimentation profitable for the owner. Moreover, when 𝑝 ∈ [𝑝, 𝑝 ̃), planning a staged offering is profitable for the owner, but launching a large IPO is not. It is also possible that planning a staged offering is profitable even if the expected market valuation of the firm's stock prior to the IPO is less than the owner's private valuation (B > E(V)).

We can summarize our results as follows:

Proposition 1. i) If it is sufficiently likely that the firm's market valuation will be low (𝑝 < 𝑝), the owner remains private; ii) If it is sufficiently likely that the firm's market valuation will be high (𝑝 > 𝑝̅ ), the owner issues a single, large IPO; iii) If uncertainty about the firms' market valuation is sufficiently high, 𝑝 ∈ (𝑝, 𝑝̅ ), the owner plans a staged equity offering, where an IPO is followed by a SEO. The SEO will be implemented only if the post-IPO market valuation is high.

Inequality (9) captures the tradeoffs of the choice between a staged offering with two smaller rounds and a single, large IPO, and inequality (10) does the same for the choice between a staged offering and staying private. Both inequalities depend on a number of parameters (p, q, 𝛿, F, B, θL and θH). From the definitions of 𝑝̅ and 𝑝 of inequalities ( 9) and ( 10) we can establish that 𝜕𝑝̅ /𝜕𝑞 ≥ 0 and 𝜕𝑝/𝜕𝑞 ≤ 0, with the inequalities being strict unless δ = 0. In words, a more revealing IPO process unambiguously makes a staged equity offering more attractive for the owner. Similarly, a straightforward differentiation of the definitions of 𝑝̅ and 𝑝 suggests that 𝜕𝑝̅ /𝜕𝛿 ≥ 0 and 𝜕𝑝//𝜕𝛿 ≤ 0, with the inequalities being strict unless q = 1/2, and that 𝜕𝑝̅ /𝜕𝐹 ≤ 0 and 𝛿𝑝̅ /𝜕𝐹 ≥ 0, with the inequalities being strict unless q = 1/2 or δ = 0. Intuitively, a stronger need for financial slack (smaller δ) or larger fixed costs of public offerings (larger F) unambiguously make a staged equity offering less attractive for the owner.5

We summarize these comparative static results as follows.

Proposition 2. A more informative IPO process (larger q), a smaller need for financial slack (larger δ)

and a smaller fixed costs of public offerings (smaller F) make planning a staged equity offering more attractive.

Implications for an empirical analysis

Our model does not (need to) take a stance on the pricing of IPOs. However, to operationalize our model, we assume that initial offering price is in the range of the lowest and highest possible post-IPO valuations (between Vl. and Vh.). For example, a fair IPO price would be E(V) ∈ (Vl., Vh). Under that assumption, our model predicts an upward (respectively, downward) price revision whenever an IPO produces the signal S = h (respectively, S = l), leading to Vh. (Vl) as the firm's post-IPO market valuation. 6The sample in our empirical analysis consists of the firms that have issued an IPO. Thus, our analysis focusses on the firms that satisfy inequality (10). Our model suggests that these IPO firms can be grouped into the following three categories: 1) Firms that plan and implement only an IPO; 2) Firms that plan and implement a staged equity offering where a SEO follows an IPO; 3) Firms that plan a staged equity offering but implement only an IPO.

In comparison to the firms in categories in 2 and 3, the firms in category 1 implement a larger IPO, have more benefits from financial slack, and are easier to value prior to an IPO. Furthermore, the firms planning a single IPO benefit less from an informative IPO process. We hence expect that the IPOs of the firms in category 1 are less informative than the IPOs of the firms in categories 2 and 3. A difference between the firms in categories 2 and 3 is that the firms in category 2 experience an upward price adjustment after the IPO, whereas the firms in category 3 have a downward price adjustment.

Data and a hard to value metric

We follow [START_REF] Hertzel | Public Market Staging: The Timing of Capital Infusions in Newly Public Firms[END_REF] and use the IPO as a common reference point for our sample firms.

We collect data on IPOs from 1980 to 2018 from Thomson SDC Platinum. Our original sample includes 19,674 IPOs. We make several restrictions to our sample: (1) We require our sample firms to have accounting data in COMPUSTAT for the fiscal year end prior to the IPO. (2) We exclude spin offs, unit offerings, and issues by financial firms with the primary standard industrial classification (SIC) code between 6000-6999. (3) We exclude firms with total assets below $10 million in 2016 dollars. (4) We further follow [START_REF] Bradley | Partial Adjustment to Public Information and IPO Underpricing[END_REF], and exclude firms with offer price below $1 per share. After these restrictions, we are left with 3,516 IPOs. The size of our final sample is comparable to other studies that make use of pre-IPO accounting variables in COMPUSTAT, such as [START_REF] Baker | Market Timing and Capital Structure[END_REF] and Alti (2006). 7We track the equity issuing behavior of the firms in our IPO sample, and consider all SEOs that occur within two years of the IPO (until the end of 2020). [START_REF] Hertzel | Public Market Staging: The Timing of Capital Infusions in Newly Public Firms[END_REF] argue that capital infusions that occur more than two years after the IPO are less likely to be sequential financing efforts that are planned at the time of the IPO. They report that 576 out of their 4054 sample IPO firms express in the original prospectus their intent to return to the capital markets, and 95% of those 576 firms indicate that they will do so within two years. As we observe SEOs that follow IPOs, we exclude SEOs during the first month of trading and consider SEOs taking place one month following the IPO. 8We use principal component analysis (PCA) to derive our metrics on difficulty to value. To identify our PCA input variables, we rely on prior literature as follows. R&D expenses is a relevant metric, as the value of research-intensive firms depends more on the future expectations arising from their growth options.

This makes them difficult to value [START_REF] Hertzel | Public Market Staging: The Timing of Capital Infusions in Newly Public Firms[END_REF]. We include firm age, as more established firms have a longer track record, which eases their valuation [START_REF] Lowry | The Variability of IPO Initial Returns[END_REF]. A greater uncertainty surrounding valuation of the firm can be reflected in the magnitude of price revisions, both negative and positive, during the IPO process [START_REF] Lowry | The Variability of IPO Initial Returns[END_REF]. This motivates us to include a measure for price revisions. Finally, firms in high tech industries and firms with negative earnings are more difficult to value [START_REF] Colak | Political Uncertainty and IPO Activity: Evidence from U.S. Gubernatorial Elections[END_REF], and we thus include indicators for both negative earnings and high tech firms. These variables each focus on different dimensions of difficulty to value a firm. As our main metric of hard to value (HTV), we use the first component (with the highest eigenvalue) from the PCA including the three of the variables mentioned above, also used by [START_REF] Gompers | Optimal Investment, Monitoring, and the Staging of Venture Capital[END_REF], namely R&D/sales, ln(Age) and the High tech indicator. Our metric explains 44.2% of the common variation amongst the three variables, and each of the three variables load with the expected sign on the first component, as follows: R&D/sales 0.6225, ln(Age) -0.3536, and High Tech 0.6982. For robustness purposes, we utilize an alternative PCA-based component (HTV2), derived from all five uncertainty-related variables namely R&D/Sales, Firm Age, an indicator for High Tech industries, negative earnings indicator, and the absolute price revision. We note that due to missing data on absolute price revision observations, the sample size decreases from 3,516 to 1,929 observations when we use HTV2.

We include several control variables in our analysis, defined in Appendix 1. In addition to standard controls explaining the IPO size and the SEO likelihood (see, e.g., [START_REF] Hertzel | Public Market Staging: The Timing of Capital Infusions in Newly Public Firms[END_REF], we include a variable (UW Premium) capturing underwriter agency costs as in [START_REF] Hoberg | The Underwriter Persistence Problem[END_REF] and [START_REF] Chang | Pre-market Trading and IPO Pricing[END_REF]. [START_REF] Chang | Pre-market Trading and IPO Pricing[END_REF] argue that underwriter agency problems are exacerbated in firms with greater capital demand, and this may drive firms to sequential issuance. Firms accept to leave money on the table in the IPO due to planned follow-on offerings. To control for underwriter-driven staged financing, we follow [START_REF] Hoberg | The Underwriter Persistence Problem[END_REF] and measure the UW Premium as the underwriter specific average underpricing. We further include an IPO market heat measure that builds on [START_REF] Boehme | Primary Market Characteristics and Secondary Market frictions of Stocks[END_REF] and [START_REF] Yung | Cycles in the IPO Market[END_REF] classification of hot and cold markets. To calculate the measure, we start with the number of quarterly IPOs, gathered from Jay Ritter's web page.9 To smooth out seasonality effects in our heat measure (4 th quarter IPOs are more likely than 1 st quarter IPOs), we calculate the measure using the average over the last four quarters. The market heat measure is the four-quarter average of the number of quarterly IPOs divided by the historical average.

We present the descriptive statistics of our sample in Table 1. Besides the full sample, Table 1 also provides information on subsamples, where we divide our full IPO sample into terciles, based on the HTV metric10 . We label the extreme terciles in Table 1 as Easy to value (ETV) and Hard to value (HTV), respectively.

As Table 1 1 deviate significantly between the HTV and the ETV group motivates us to consider the differences in a regression setting.

< Table 1 about here>

In Panel A of Table 2, we test whether our pre-IPO hard to value metric is linked to more widely used post-IPO measures of valuation uncertainty, namely idiosyncratic volatility (see, e.g. [START_REF] Jiang | Information Uncertainty and Expected Returns[END_REF][START_REF] Kumar | Hard-to-Value Stocks, Behavioral Biases, and Informed Trading[END_REF][START_REF] Zhang | Information Uncertainty and Stock Returns[END_REF] and volatility. In panel A of Table 2, we sort the firms into terciles based on HTV, and report that HTV firms have significantly higher idiosyncratic volatility and volatility in both 1-month and 3-month periods following the IPO. In unreported sortings, HTV2 yields similar inferences.

This supports validity of HTV as a measure of valuation uncertainty.

Following proposition 1, our main hypothesis is that difficulty to value is negatively related to IPO size and positively related to the SEO likelihood. The increased SEO likelihood comes from two channels.

Firstly, HTV firms conduct smaller IPOs, which should lead them to return to the equity market soon.

However, conducting a small IPO is not sufficient to trigger a second round of equity issuance following the IPO. The small IPO needs to be coupled with a positive stock market return. Panel B of Table 2 considers two of those implications in a setting where we sort IPO firms both on our hard to value metric and on post-IPO return, calculated for the one-month period following the IPO. We group both the HTV metric and the returns into terciles, and compare the likelihood of an SEO within two years between the extreme terciles.

Panel B of Table 2 indicates that difficulty to value the firm has a significant effect on SEO likelihood, regardless of post-IPO return. While low return firms are less likely to issue an SEO within two years, even among low return firms, the hard to value ones are more likely to issue a follow-on SEO.

Similarly, post-IPO returns have a positive effect on SEO likelihood in all difficult to value terciles. SEO likelihood increases across both HTV terciles and alpha terciles. Both findings are consistent with the predictions of our model.

<Table 2 about here>

Regression results

In this section, we test the relation between HTV, IPO Size and post-IPO equity issuance in a regression setting. As we note above, our expectation is that firms with uncertainty surrounding their valuation conduct smaller IPOs and then engage in sequential equity financing, and further that firms in which the small IPO is followed by positive returns tend to return to the equity market shortly after their IPO.

We begin our regression analysis by testing whether IPOs of difficult to value firms are smaller than those issued by easy to value firms, as smaller IPO size may indicate that only a portion of total equity needs is raised at the IPO stage. In the first two columns of Table 3, we run OLS regressions where the dependent variable is the IPO size (Shares offered in the IPO/Post-IPO shares outstanding). We employ year fixed effects to control for the annual variation in macroeconomic factors and market conditions. To further test whether our results regarding the effects of HTV on the IPO Size also hold within industries, we include industry fixed effects in Column (2). We add controls for profitability [Returns on Assets (ROA)], size [ln(Sales)], indebtness (book leverage), and Cash/Assets, all measured at the fiscal year end prior to the IPO, and market to book at issuance. As issuance-specific controls, we further include IPO Heat, VC Backing, Top Underwriter and Underwriter Premium. The HTV metric exhibits a strong negative relation to IPO Size, suggesting that difficult to value firms issue IPOs with smaller relative proceeds.

Results on our hard to value proxy are unaffected by inclusion of industry fixed effects in Column (2).

Interestingly, cash holdings are inversely related to IPO Size in both specifications. This is consistent with the idea that firms with a greater degree of financial slack conduct a smaller IPO.

<Table 3 about here> Next, we consider the determinants of the likelihood to return to the equity issuance market within two years of the IPO by running regressions where the dependent variable is a binary variable that takes the value of one for those IPO firms that conduct a follow-on SEO within two years of the IPO. In this setting, our model predicts that our HTV measures are positively related to the SEO likelihood. We estimate with OLS instead of Probit due to the large amount of fixed effects when we include industry fixed effects, and the resulting inconsistency in the estimates. 11 The structure of Columns ( 3) and ( 4) follows those of Columns ( 1) and (2) of Table 3, but further add several post-IPO controls to capture the firm's behavior during the first month of trading [First day return, 21-day return, volatility and ln(Analyst)], as mentioned below. 12If information production is essential in allowing the second stage follow-on SEO, then the issuing firm would have an incentive to improve information production during the IPO process, and immediately after it. For this reason, we include the number of analysts following the firm as a control for informativeness. Prior studies suggest two other variables that can function as pre-certification devices, by improving information production of a newly-listed firm. [START_REF] Megginson | Venture Capitalist Certification in Initial Public Offerings[END_REF] and Megginson, et al. (2017) report that VC Backing of the IPO enhances information dissemination during the IPO process.

Francis and Hasan (2001), and Francis, et al. (2010) forward the "analyst coverage purchase hypothesis", which posits that firms engage top-ranked underwriters to improve information production in the secondary market. [START_REF] Chen | Media Coverage and IPO Pricing around the World[END_REF] provide support for the hypothesis that intermediaries and venture capitalists improve information production in IPOs. In untabulated tests, we find that firms that are more difficult to value (according to our HTV measures) are significantly more likely both to have VC Backing and to employ Top Underwriters, defined as in [START_REF] Loughran | Why has Underpricing Changed Over Time?[END_REF].

HTV enters with a positive and significant coefficient, both in Column (3) with the year fixed effects, and in Column (4) with year and industry fixed effects. In line with the implications of our model, we find that IPO Size is negatively related to the SEO likelihood.

Furthermore, in contrast to [START_REF] Chang | Pre-market Trading and IPO Pricing[END_REF] argument that high underwriter premium is linked to a greater likelihood of a follow on offering, UW premium fails to reach conventional levels of statistical significance in our specifications. Together, our tests support our model's implication that firms that are difficult to value use sequential equity financing, as they both issue smaller IPOs, and have a greater likelihood of returning to the equity market within two years.

To test the severity of a potential omitted variable bias, we employ the partial identification methodology by [START_REF] Oster | Unobservable Selection and Coefficient Stability: Theory and Evidence[END_REF]. In the test, we create bounds for the betas of HTV in specifications (1) and

(3) of Table 3. In unreported estimates, the intervals of the HTV betas is given by (R 2 max = 1.3Ř 2 and delta =1) and the controlled betas where Ř 2 is the R-square from the model including controls. Our estimates indicate that no zeros exist within the intervals when using the solution that minimizes the distance to the controlled estimate. Hence, the observables in our model are likely to be at least as important as the unobservables in determining the IPO size and SEO likelihood.

In Columns ( 5) and ( 6) of Table 3, we consider the connection between sequential equity financing and difficulty to value in yet another setting. [START_REF] Hertzel | Public Market Staging: The Timing of Capital Infusions in Newly Public Firms[END_REF] motivate studying sequential financing behaviour with a Cox hazard model, where the dependent variable is the time period between the IPO and the subsequent capital injection. For this analysis, we observe equity issuance of the IPO firms for the two years or 730 days following the IPO, so for those firms that do not issue a follow on SEO, the dependent variable receives its maximum value of two years.

The results of the Cox hazard model are consistent with those reported in Columns (3) and ( 4). The hazard rate exhibits a strong connection to our key variables, so that difficulty to value, IPO size and post-IPO returns have a strong effect in shortening the time between the IPO and the SEO, and IPO size lengthens the time to first SEO. Some of our control variables provide contrasting inferences between specifications.

In both specifications, First day return, ln(analyst), top underwriter, and leverage shortens and IPO market heat, profitability and market-to-book lengthens the time between the IPO and first SEO.

As a robustness exercise, we re-estimate the models of Table 3 using the five-variable HTV2 metric. The sample size decreases from 3,516 to 1,929 observations due to missing observations in the absolute price revision variable. Our findings in Table IA2 in the internet appendix remain qualitatively similar to the ones in Table 3. Again, our findings suggest that the hard to value metric is linked to smaller IPOs, increased SEO likelihood and a shorter duration to the first SEO.

Staged financing that follows an IPO can take forms other than a follow-on SEO. Among the 1,922 firms that [START_REF] Hertzel | Public Market Staging: The Timing of Capital Infusions in Newly Public Firms[END_REF] observe to raise public financing within two years of their IPOs, 52% issue debt within two years, and about 48% follow with an SEO. In our analysis, we focus on sequential equity financing, as our model does not provide predictions related to other forms of financing. However, in Table IA1 (internet appendix) we test the effect of our HTV proxies on large debt increases (>5%) occurring within two years of the IPO. We report a negative relation between HTV and debt increases in one of the specifications. The other three specifications exhibit effects close to zero. Our findings support the notion that debt and equity issuance have different sensitivities to information uncertainty. However, as both cash holdings and ROA negatively relate to debt issuance, debt issuance is also an important source for post-IPO firms to fund financing deficits.

Next, we study the joint effects implied by our model. In Table 4, we focus on the triple interactions between post-IPO returns, HTV metrics, and IPO Size. In Columns ( 1) and ( 2) of Table 4, we interact small IPO Size with high one-month returns and a hard to value indicator (top tercile HTV1) with the SEO likelihood as the dependent variable. The positive and statistically significant coefficients for the triple interaction variable suggest that the likelihood of a follow-on SEO is elevated for firms with small IPOs, high post-IPO returns, and for firms that they are difficult to value. In Columns (3) and (4), we repeat our analysis focusing on the time to the SEO, using a Cox hazard model. Consistent with the predictions of our model, we find a triple interaction between IPO Size (conducting a small IPO), returns and HTV to have a positive effect on the likelihood of a follow-on issue.

<Table 4 about here>

Further analysis and alternative explanations

This section begins with an analysis of whether our HTV proxy captures valuation uncertainty or asymmetric information. Second, we consider the potential selection bias present in empirical tests. Third, we test whether our results can be explained by alternative explanations of sequential equity financing.

Fourth, we study the role of the IPO in the two-sided learning process about firm valuation.

Asymmetric or imperfect information

In an attempt to separate between asymmetric and imperfect but symmetric information structures, we analyze the insiders' participation in the IPO. By studying the insiders' actions we can gain understanding of the informational wedge between the insiders and the outsiders [START_REF] Chen | Price Informativeness and Investment Sensitivity to Stock Price[END_REF]Focault Fresard, 2014). In an asymmetric information structure, insiders are better informed than outsiders, and can thereby make better-informed decisions. In the IPO, this can be reflected for instance in market timing of their secondary share sales. If insiders act on asymmetric information, their sales of IPO shares should be positively related to valuation uncertainty (or degree of asymmetric information) and negatively related to post-IPO returns. 13 In a symmetric but incomplete information structure, the value of the option to sell shares at a later stage should increase with uncertainty, causing either a staged offering of secondary shares or postponing of secondary shares issues until information is more complete (see, e.g., Damaraju, et al., 2015). Hence, we expect that participation ratio and secondary share sales decrease with valuation uncertainty. We conduct two different tests: 1.) We analyze the link between the HTV proxy and the sale of secondary shares in the IPO. 2.) We acknowledge that in asymmetric information structures, insiders can better forecast future returns and test the link between insider participation and post-IPO performance. If insiders are better informed about the firm's future prospects, they will sell a larger fraction of shares in the IPO given that the expected future returns are low. 14 In our tests, the participation ratio is defined as the fraction of secondary shares offered in the IPO (Chanine, et al., 2020). We also include a second measure 13 The role of asymmetric information and market timing behaviour is prevalent in the literature. For example, [START_REF] Bergstrasser | CEO Incentives and Earnings Management[END_REF] find that management manipulates earnings prior to share sales with a resulting post sell off decrease in stock prices. Also, Bartov and Mohanran (2004) report that firm's share price drops following employee stock option exercises. However, in the IPO setting, it is plausible to argue that even insiders do not know the true valuation of their firm prior to the listing (See, e.g., [START_REF] Brau | Initial Public Offerings: An Analysis of Theory and Practice[END_REF]. 14 E.g. [START_REF] Jordan | Volatility and Mutual Fund Manager Skill[END_REF] use a similar framework with alpha as the left hand side variable.

of secondary share sales defined as secondary shares scaled by total post-IPO shares outstanding (Secondary/Total).

<Table 5 about here> Table 5 reports our findings on secondary share sale. In Columns ( 1) and ( 2), we study the effect of the HTV measure on the participation ratio and Secondary/Total measures. In both Columns, we report a negative relation to the HTV proxy. In Columns (3) to (6) of Table 5, we study the effect of insider participation in the IPO on post-IPO performance, measured by the 1-month and 6-month return. At the month horizon, the participation ratio is positively related to future returns. We fail to find consistent evidence of a link between insider participation and higher post-IPO stock market performance in the other three specifications. 15 To summarize, our results fail to support the notion that insiders take advantage of an asymmetric information structure to maximize their own profits. Even though we cannot completely rule out the effect of information asymmetry on our results, our evidence suggests that insiders act upon uncertain information and postpone issuing their own shares. As further evidence against the insider market timing explanation, future returns are not inversely related to insider IPO participation. Our results are consistent with [START_REF] Mclean | Share issuance and cash savings[END_REF], who studies cash savings from IPO issues and finds that firms are increasingly precautionary in use of their IPO proceeds and that this trend is unaffected by market timing.

<Table 6 about here>

In Table 6 we study the robustness of our HTV proxy against commonly used measures of asymmetric information. In both the IPO size and SEO likelihood regressions, we include the Amihud (2002) illiquidity measure, idiosyncratic volatility and volatility as right-hand-side variables (all measured up to one month following the IPO). Since, it is possible that the post-IPO measures convey information not contained in our proxy, we include the post-IPO variables as controls in the IPO size regressions. Our findings remain intact following the inclusion of the post-IPO asymmetric information measures in both specifications.

Selection bias

Our model predicts that some firms will stay out of the market, creating an unobservable counterfactual that is a potential source of selection bias. We have three potential selection effects to consider. First, our model predicts that firms with a low value of p do not enter the market. Second, our HTV proxy fails to account for the role of cross peer learning in the IPO decision. Third, firms with venture capital backing can use the reputation of the VC firm to go public.

It is unavoidable that some bias arises on HTV, stemming from the self-selection to conduct an IPO. ETV firms with a low p effectively stay out of the market. In this case, the observed quality of ETV firms in our sample is likely to be upward biased. This reduces quality heterogeneity among ETV firms relative to HTV firms. [START_REF] Booth | Capital Raising, Underwriting and the Certification Hypothesis[END_REF]Megginson, et al., 2017). Also IPO heat can be linked to quality, as [START_REF] Khanna | Good IPOs draw in bad: Inelastic banking capacity and hot markets[END_REF] and [START_REF] Yung | Cycles in the IPO Market[END_REF] conclude that firm quality is greater during cold IPO markets.

The matching of ETV and HTV firms enables us to capture firms of comparable observable quality. We match the top tercile of HTV firms with the two lower terciles of HTV using propensity scores, we use a caliper of 0.01 and no-replacement. Due to the large differences between HTV firms and the control group, we end up with 1,678 observations. Results in Table 7 Columns ( 1) and ( 2) provide continued support for a negative impact of both HTV proxies on IPO size. Columns (3) and ( 4) shows that the SEO likelihood remains positively affected by the HTV proxy.

<Table 7 about here> A second selection concern comes from the possibility that uncertainty concerning the value of p (reflected in q in our model) is affected by external factors such as peer group valuation levels and their information content. Prior literature establishes that firms can obtain a greater precision in their valuation by learning from peer firms that are already listed (Agahamolla and Thakor, 2020;[START_REF] Foucault | Learning From Peers' Stock Prices and Corporate Investment[END_REF][START_REF] Grennan | Dividend Payments as a Response to Peer Influence[END_REF][START_REF] Kaustia | Social Learning and Corporate Peer Effects[END_REF]. 16 To test for this case of learning, we create peer groups of listed firms based on [START_REF] Hoberg | Product Market Synergies and Competition in Mergers and Acquisitions: A Text-Based Analysis[END_REF]2016) in a selection bias, where high uncertainty firms can only go public when their peer valuations are high and informative. 17 The moderating role played by the precision and level of the peer group valuations suggests that our estimates could be in the lower bound.

To further study the effect of peer group information on sequential equity financing and thus robustness of our main findings, we add the peer group variables as controls. Due to the high correlation between peer group information and our HTV proxy 18 we cancel out the peer group effect present in the 16 In related theoretical work, [START_REF] Benveniste | Information Externalities and the Role of Underwriters in Primary Equity Markets[END_REF] argue for learning from recent peer firm IPOs. [START_REF] Benveniste | Evidence of Information Spillovers in the Production of Investment Banking Services[END_REF], find empirical evidence of firms from the same industry going public on similar terms (common valuation factor). 17 A possible channel keeping high uncertainty firms out of the IPO market is that the elevated peer group uncertainty maps into higher cost of capital causing low valuation expectations (see, e.g., [START_REF] Colak | Political Uncertainty and IPO Activity: Evidence from U.S. Gubernatorial Elections[END_REF]. 18 HTV has a correlation coefficient of 0.24 with average peer group M/B and 0.38 with the number of peer group analysts.

HTV proxy by orthogonalizing that proxy with the peer group M/B and peer group analysts per firm.

Orthogonalizing the variables allows us to study the individual valuation uncertainty on IPO size and SEO likelihood. 19 This approach provides us with an indication of the direction of the bias. Using the [START_REF] Hoberg | Product Market Synergies and Competition in Mergers and Acquisitions: A Text-Based Analysis[END_REF]2016) peer groups significantly reduces our sample size from 3,625 to 2,615, since the data is only available from 1989 onwards. We therefore re-estimate the base case models for comparability.

Columns ( 1) and ( 2) in Panel B of Table 8 show the baseline estimates using the reduced sample.

In line with prior estimates, HTV has a negative impact on IPO size and a positive impact on the SEO likelihood. When we add the peer group controls in Columns ( 3) and ( 4), the relation between HTV and IPO size strengthens, but the finding on SEO-likelihood (in Column 4) is no longer statistically significant.

Orthogonalizing the measures in Columns ( 5) and ( 6) strengthens the relationship between HTV and both IPO size and SEO likelihood. In our setting, we cannot observe the counterfactual of firms not conducting an IPO. Therefore, we can only make predictions concerning the selection bias. In an attempt to bound the bias, we control for available peer information in a regression setting and report an absolute increase in the coefficient estimates. 20 After taking into account one source of potential selection bias, we can observe that our HTV estimates are not downward biased.

<Table 8 about here> Finally, a third source of IPO-selection can potentially stem from venture capital backed IPOs.

Venture capital firms often invest in young and R&D intensive firms operating in the tech sector. As shown in Table 1, 72% of the HTV firms are VC backed, compared to only 15.5% in the ETV sample. Venture capitalists usually sell most of their portfolio companies through trade sales and publicly list only the highest quality ones [START_REF] Bayar | IPOs versus Acquisitions and the Valuation Premium Puzzle: A Theory of Exit Choice by Entrepreneurs and Venture Capitalists[END_REF]2012). To verify that our finding is not purely driven by venture capital backed firms, we exclude all venture capital backed IPO firms from our sample and reestimate the models of Table 3. The estimations of Table 9 show that our findings are robust to excluding 19 Orthogonalization of the main independent variable is used in other studies to cancel out a selection effect [START_REF] Bhattacharya | Firm Mortality and Natal Financial Care[END_REF] or to extract private from public information (Aggarwal and Hauswald, 2020). 20 Grosse-Rueschkamp et al. ( 2019) use a similar approach to bound the selection bias.

venture capital backed firms, HTV has a negative impact on IPO size and the time to the SEO and a positive impact on the SEO likelihood.

<Table 9 about here>

Post-IPO returns and alternative explanations

In our next set of empirical tests, we pursue alternative explanations behind our results. First, firms may time the market for their subsequent SEO issuance, and such timing could correlate with our measures of valuation uncertainty. While we use year fixed effects in our tests to account for market-wide patterns, market timing on firm-level information is a possibility. Our model predicts that positive post-IPO returns trigger firms to conduct an SEO, but seasoned equity offerings that follow high returns can also be driven by market timing [START_REF] Baker | Market Timing and Capital Structure[END_REF][START_REF] Taggart | A Model of Corporate Financing Decisions[END_REF], which is thus another possible explanation to the findings we report above. However, market timing would also imply that the SEO is followed by low returns, as firms optimize to take advantage of high valuations. In Table 10, we report the pre-and post-SEO Alphas, along with the coefficients for the factors of the Fama-French five-factor model, for calendartime portfolios that are formed based on terciles of valuation uncertainty. The first three columns study the return patterns from IPO to SEO, while the remaining three columns show the post-SEO return estimations.

The sample includes firms that follow their IPOs with SEOs within two years. If the main motive for SEO issuance were market timing based on firm-specific information, the alphas should be negative in the period following the SEO. Our findings in columns (1) to (3) of Table 10 show that pre-SEO alphas are highest for high valuation uncertainty firms. While this is in line with our model, it could also indicate that managers just time the market. Therefore, we analyze the post-SEO returns in columns (4) to ( 6). Contrary to a market timing explanation, post-SEO alphas are strongly positive, which suggests that market timing is not a major consideration when firms make their decision to float a follow-on SEO.

<Table 10 about here>

As our model predicts that returns subsequent to the IPO trigger an SEO, we next study the impact of the first 12-months return on the SEO likelihood. Our findings in Table 11 show a decreasing sensitivity of SEO likelihood to returns over time with two exceptions. First, the SEO likelihood is the most sensitive to returns during month 6. A potential explanation is that lock-ups normally expire after 180 days and insiders may sell secondary shares through an SEO. Second, month 3 exhibits the third highest sensitivity after month 6 and month 1. The benefits from two-way learning should occur predominantly quickly after the IPO. Results in Table 11 provide support for this notion.

<Table 11 about here> Another alternative explanation to our findings could be investors' risk aversion and resulting unwillingness to participate in IPOs of hard to value firms, which would force firms to conduct smaller IPOs and later return to the equity markets for an IPO. To explore this explanation, we interact hard to value with two measures of risk aversion. First, we use the Zhou (2018) variance risk premia 21 . Second, we use the [START_REF] Baker | Investor Sentiment and the Cross-Section of Stock Returns[END_REF] sentiment index 22 . We interact the highest (lowest) tercile of variance risk premium (market sentiment) with the top tercile of the HTV proxy. Since, both variance risk premium and market sentiment are time-series variables, we exclude year fixed effects from the estimations. To control for macro-economic heterogeneity we include U.S. GDP growth and the fed funds rate. Our findings in Table 12 suggest that hard to value firms are not disproportionally affected by investors´ risk aversion.

<Table 12 about here> Next, we pursue a motive for sequential equity financing suggested by the literature on underpricing and sequential offerings (See, e.g., [START_REF] Allen | Signaling by Underpricing in the IPO Market[END_REF][START_REF] Jegadeesh | An Empirical Investigation of IPO Returns and Subsequent Equity Offerings[END_REF][START_REF] Welch | Seasoned Equity Offerings, Imitation Costs, and the Underpricing of Initial Public Offerings[END_REF]. In an asymmetric information setting, firms may signal their quality through underpricing. High quality firms know that their valuation will increase at a later stage, and therefore conduct a smaller IPO and at a later stage issue equity at a higher valuation. As the HTV proxy is positively correlated with both underpricing and post-IPO asymmetric information measures, it is possible that our findings are driven by staged equity 21 The data is retrieved from Hao Zhou's webpage: https://sites.google.com/site/haozhouspersonalhomepage 22 The data is retrieved from Jeffrey Wurgler's webpage: http://people.stern.nyu.edu/jwurgler/ financing determined by asymmetric information. To control for the underpricing signaling channel of staged financing, we include underpricing in the IPO size regressions as in [START_REF] Boone | Redacting Proprietary Information at the Initial Public Offering[END_REF]. Column (1)

in Table 13 shows that the relation between HTV and IPO size remains intact after the inclusion of first day return as a right hand side variable.

<Table 13 about here> Finally, equity financing may occur to finance long-term cash flow deficits [START_REF] Huang | Corporate Cash Shortfalls and Financing Decisions[END_REF]. [START_REF] Denis | Persistent Negative Cash Flows, Staged Financing, and the Stockpiling of Cash Balances[END_REF] find that cash flow deficits play a role in staged equity financing. They further document that firms raising cash through frequent equity issues are R&D intensive firms within the tech and health care sectors. Hence, it likely that our HTV measures correlate with negative cash flow firms. Next, we test if the correlation between the HTV metric and firms' cash needs is driving our results.

To measure cash needs, we follow [START_REF] Denis | Persistent Negative Cash Flows, Staged Financing, and the Stockpiling of Cash Balances[END_REF] and include a net cash flow variable and an indicator for negative cash flows together with an interaction term. We measure net cash flows as cash flows from operations plus cash flows from investment activities. Column (2) of Table 13 indicates that firms with higher cash flows conduct larger IPOs. However, our findings in Column (3) fail to support a connection between negative cash flows and SEO likelihood. Our HTV proxy remains a significant determinant for IPO size and SEO likelihood after controlling for negative cash flows and first day returns.

The role of learning in the IPO process

Our HTV proxy uses information from the period prior to the IPO process. Theoretically, we can decompose valuation uncertainty into three components. First, the symmetric but imperfect information that is known to both insiders and outsiders of the firm. Second, the proprietary information of the management, creating information asymmetries for outsiders. Third, following [START_REF] Benveniste | How Investment Bankers Determine the Offer Price and Allocation[END_REF], the IPO process can be seen as a partial learning channel reflecting q in our model, as outsiders potentially possess information to aid management in learning the true valuation of their firm. During the IPO process, the S-1 filings and the following road shows are in place to reduce management's private information of the firm's prospects. The road shows also offer two-sided learning, where not only information asymmetries are reduced but also management learns about firm valuation from outsiders.

However, in our model we assume that the information asymmetry is zero at the time of the IPO. Even though, recent evidence suggest that the S-1 filings already provide substantial information of the IPO, even above and beyond the value of road shows [START_REF] Blankespoor | The Pitch: Managers' Disclosure Choice during IPO Roadshows[END_REF]. The informativeness of the S-1 filings also varies across IPOs [START_REF] Boone | Redacting Proprietary Information at the Initial Public Offering[END_REF][START_REF] Hanley | The Information Content of IPO Prospectuses[END_REF].

In order to study the effect pre-IPO information and IPO process learning on IPO size and SEO likelihood, we include absolute price revision in the regression along with gross spread, first day return and Post-IPO analyst following, as these variables should capture information about the IPO process uncertainty [START_REF] Chen | The Seven Percent Solution[END_REF]. By separating pre-IPO process and IPO process uncertainty, we are able to study the importance of sequential learning occurring in the IPO process.

<Table 14 about here>

The results in Table 14 align with our previous findings. HTV has a negative impact on IPO size and a positive impact on SEO likelihood. We report that gross spread is negatively related to IPO size and that post-IPO analyst following also correlates positively with the SEO likelihood. In line with [START_REF] Loughran | Why don't Issuers Get Upset About Leaving Money on The Table in IPOs?[END_REF], and [START_REF] Lowry | Is the IPO Pricing Process Efficient?[END_REF], our findings suggest partial learning during the IPO process. However, the remaining explanatory power of variables from the S-1 filings and IPO size in determining the SEO likelihood suggests that such learning is incomplete. The incomplete information production in the IPO process further incentivizes managers who are uncertain about their pre-IPO valuations to stage their firms' equity issuance.

Conclusions

We examine the impact of valuation uncertainty on staged equity financing. We seek to shed light on why a number firms choose to return to the equity market shortly after their IPOs. We develop a simple model that indicates that under valuation uncertainty, firms may stage their equity issuance. We obtain predictions that hard to value firms choose to conduct smaller IPOs and that they thereafter return to the equity if they experience a positive post-IPO return. We test these predictions using a sample of 3,625 U.S.

IPOs between 1.1.1980-31.12.2018. Our results provide support for our model, as we find a strong effect of both post-IPO alphas and firm level valuation uncertainty on the likelihood that the firm returns to the equity markets. New to the literature, we provide both a simple rationale and empirical support for the idea that information asymmetry is not a necessary criterion for raising equity financing in stages.

As our contribution is mainly on the empirical side, our theoretical model leaves open many exiting questions. For example, we have been agnostic as to the owner's use of the funds raised from a public listing. A future work should consider the owner's choice of whether to go public so as to cash out or to raise funds for a real investment project affecting the firm value. The net present value of the project could depend on its size and there could be learning about the value after the IPO. The table shows sorting on the difficulty to value (HTV). We measure the difficulty to value with the first principal component from R&D scaled by sales, firm age and a high tech indictor. ETV denotes easy to value, Mid is the mid tercile and HTV is the most difficult to value tercile. Panel A reports the 1-month idiosyncratic volatility from a Fama-French 5-factor model (IVOL) and 1-month return volatility (VOL) over the three terciles. Panel B reports the SEO likelihood between 1-24 months following the IPO. In addition to sorting on valuation uncertainty, we further sort on the firms' one month return after the IPO. ***, **,* denote 1%, 5%, 10% significance, respectively. 2) and SEO likelihood (SEO 1/0) in column (3). The main independent variable is the HTV proxy, first day return, net cash flows, a negative net cash flow indicator and the interaction between net cash flows and the net cash flow indicator. Models (1) and ( 2) include the controls of column (1) of

  indicates, ETV firms deviate from HTV firms in a number of aspects. Deviations in, ln(Age), R&D/Sales and Hi Tech, are to be expected, as those variables form the basis for the firm groupings. With respect to our model of sequential financing with equity, the most interesting statistics in Table1are related to subsequent SEO issuance and IPO size. Consistent with our expectations, hard to value firms are significantly more likely to issue follow-on SEOs within two years (43.6% vs. 31.5%). Hard to value firms also float smaller IPOs, which aligns well with the predictions generated by our model. The fact that means of all of the variables listed in Table

  peer group classification, which builds on product market similarity scores. A peer group is assigned to an IPO firm if they are included in the Hoberg and Phillips data within the two years following the IPO. We create two peer group measures at the time of the IPO. First, we observe the valuation of the peer group, measured by the average M/B, as defined by[START_REF] Foucault | Learning From Peers' Stock Prices and Corporate Investment[END_REF]. Second, we create a measure of informativeness of the peers' valuations through aggregating the peer group's analyst following. We then sort firms on peer group valuation and peer group informativeness.

<Table 8 about here> Our findings in Panel A of Table 8 points towards a strong selection effect. Firms that are easier to value [(HTV) is low] have low valuations and less informative peers. The outcome from the sortings suggests that cross-peer learning can play a moderating role on individual firm uncertainty. Low and uninformative prospective valuations effectively keep high uncertainty firms out of the market. This results

Table 1 :

 1 Descriptive statisticsThe table below reports descriptive statistics for our sample firms. Our sample consists of 3,525 U.S. IPOs between1.1.1980-31.12.2018. We sort the firms on difficulty to value (HTV). We measure the difficulty to value with the first principal component from R&D scaled by sales, firm age and a high tech indictor. For variable definitions see appendix 1. ***, **,* denote 1%, 5%, 10% significance, respectively.

			Full Sample		Easy to Value		Mid	Hard to Value
		Mean	Std. Dev.	Min	Max	Mean	Std. Dev.	Mean	Std. Dev.	Mean	Std. Dev.	t/z-stat ETV -HTV
	IPO Size	0.325	0.188	0.083	1.000	0.375	0.200	0.322	0.183	0.277	0.166	0.10***
	SEO 1/0	0.361	0.480	0.000	1.000	0.315	0.465	0.333	0.472	0.436	0.496	-0.12***
	Participation Ratio 0.154	0.224	0.000	1.000	0.207	0.260	0.131	0.202	0.124	0.196	0.08***
	Secondary/Total	0.054	0.100	0.000	1.000	0.079	0.126	0.045	0.088	0.038	0.074	0.04***
	R&D	4.816	10.327	0.000	65.104	0.365	1.257	1.739	2.972	12.436	14.985	-12.07***
	Missing R&D	0.367	0.482	0.000	1.000	0.593	0.491	0.448	0.497	0.059	0.235	0.53***
	Ln(Age)	2.484	0.890	0.693	4.635	3.336	0.658	1.871	0.569	2.249	0.678	1.09***
	Abs(Revision)	0.134	0.112	0.000	0.500	0.115	0.096	0.147	0.121	0.144	0.118	-0.03***
	Earnings <0	0.373	0.484	0.000	1.000	0.164	0.370	0.389	0.488	0.567	0.496	-0.40***
	Hi Tech	0.261	0.439	0.000	1.000	0.000	0.000	0.000	0.000	0.788	0.409	-0.79***
	ROA	0.048	0.324	-1.187	0.609	0.189	0.141	0.048	0.315	-0.093	0.396	0.28***
	M/B	2.879	2.182	0.773	13.669	2.022	1.237	3.153	2.556	3.466	2.250	-1.44***
	Ln(Sales)	4.282	1.670	-0.737	8.359	5.224	1.359	3.992	1.475	3.627	1.718	1.60***
	Leverage	0.339	0.301	0.000	1.450	0.438	0.287	0.333	0.291	0.246	0.293	0.19***
	Cash Holdings	0.202	0.245	0.000	0.993	0.071	0.103	0.190	0.225	0.346	0.286	-0.27***
	IPO Heat	1.514	0.780	0.102	3.710	1.635	0.815	1.583	0.748	1.323	0.740	0.31***
	Top Underwriter	0.585	0.493	0.000	1.000	0.543	0.498	0.590	0.492	0.624	0.485	-0.08***
	UW Premium	0.214	0.162	-0.028	0.870	0.174	0.127	0.226	0.181	0.243	0.167	-0.07***
	VC Backing	0.464	0.499	0.000	1.000	0.155	0.362	0.506	0.500	0.734	0.442	-0.58***
	Ln(Analyst)	1.029	0.513	0.000	2.833	0.946	0.530	1.003	0.489	1.138	0.499	-0.19***
	First Day Return	0.184	0.344	-0.170	2.021	0.088	0.169	0.239	0.408	0.225	0.383	-0.14***
	1-Month Return	0.031	0.198	-0.404	0.841	0.010	0.140	0.031	0.213	0.053	0.228	-0.04***
	1-Month Std Dev	0.039	0.024	0.008	0.134	0.029	0.016	0.043	0.028	0.044	0.024	-0.01***
	N	3,516				1,172		1,182		1,162

Table 2 : Average SEO likelihood sorted by post-IPO alpha and difficulty to value

 2

Table 4 :

 4 Interaction modelsThe table below shows the interactions between IPO Size, post-IPO alpha and valuation uncertainty. We include indicators if the IPO firm belongs to the bottom tercile of IPO Size, top tercile of 1-month post-IPO return and the top tercile of the hard to value PCA (HTV). All variables are defined as in appendix 1. Reported t-stats in parentheses are heteroscedasticity robust and double clustered on industry (SIC-3) and year. ***, **,* denote 1%, 5%, 10% significance, respectively.

	(1)	(2)	(3)	(4)
	SEO 1/0	SEO 1/0	Time to SEO Time to SEO

Table 5 :

 5 Secondary sharesThe table below reports regressions using secondary share sales in the IPO (models 1 and 2) and post IPO-returns (3-6) as dependent variables. All variables are defined as in appendix A1. The models include year fixed effects. Reported t-stats in parentheses are heteroscedasticity robust and double clustered on industry (FF-12) and year. ***, **,* denote 1%, 5%, 10% significance, respectively.

		(1)	(2)	(3)	(4)	(5)	(6)
		Participation	Secondary/Total
		Ratio	shares	1-Month Return	6-Months Return
	HTV	-0.009*	-0.006**	0.012**	0.012**	0.011	0.010
		(-2.20)	(-2.94)	(2.79)	(2.70)	(0.84)	(0.78)
	Participation ratio			0.037**		0.050
				(2.95)		(1.74)
	Secondary/Total shares				0.011		-0.043
					(0.39)		(-0.46)
	Leverage	-0.119***	-0.039***	-0.019	-0.023	-0.029	-0.037
		(-5.19)	(-3.38)	(-1.47)	(-1.80)	(-0.70)	(-0.88)
	ln(Sales)	0.027***	0.008***	-0.001	-0.000	-0.012	-0.010
		(3.93)	(3.33)	(-0.39)	(-0.02)	(-1.32)	(-1.16)
	M/B	0.012*	-0.001	-0.014*** -0.013*** -0.042*** -0.041***
		(2.07)	(-0.63)	(-4.56)	(-4.50)	(-4.21)	(-3.92)
	Cash/Assets	0.004	-0.003	-0.024	-0.024	0.027	0.027
		(0.13)	(-0.27)	(-1.79)	(-1.81)	(0.48)	(0.47)
	VCB	-0.004	-0.002	0.012	0.011	-0.020	-0.020
		(-0.53)	(-0.44)	(1.29)	(1.28)	(-1.43)	(-1.45)
	ROA	0.120***	0.031**	0.030	0.034*	0.049	0.057
		(4.10)	(3.16)	(1.66)	(1.82)	(0.92)	(1.05)
	IPO Heat	-0.015	-0.002	-0.006	-0.007	-0.077	-0.078
		(-0.73)	(-0.32)	(-0.38)	(-0.42)	(-1.55)	(-1.56)
	Top Underwriter	0.029**	0.010	-0.007	-0.006	0.003	0.005
		(2.36)	(1.67)	(-0.73)	(-0.64)	(0.08)	(0.13)
	UW Premium	-0.024	-0.008	-0.035**	-0.035**	0.084	0.083
		(-0.81)	(-0.71)	(-2.48)	(-2.59)	(1.71)	(1.69)
	Ln(Analyst)			0.037*** 0.038*** 0.122*** 0.122***
				(6.84)	(7.05)	(4.61)	(4.59)
	First Day Return			0.007	0.005	0.011	0.008
				(0.21)	(0.15)	(0.26)	(0.18)
	1-Month Std Dev			2.050**	2.039**	2.206**	2.186**
				(2.83)	(2.81)	(2.73)	(2.71)
	Constant	0.050	0.033**	-0.025	-0.023	0.121	0.125
		(0.98)	(2.47)	(-0.49)	(-0.47)	(1.44)	(1.47)
	Observations	3,514	3,514	3,514	3,514	3,514	3,514
	R-squared	0.218	0.127	0.093	0.092	0.124	0.124
	Year FE	Y	Y	Y	Y	Y	Y
	Industry FE	N	N	N	N	N	N

Table 7 :

 7 Matched sampleThe table below reports regressions using a matched sample based on propensity scores. IPO size is the dependent variable in columns (1) and (2) and an indicator SEO 1/0 if the firm conducts an SEO within two years following the IPO in columns (3) and (4). All variables are defined in appendix A1. Reported t-stats in parentheses are

	heteroscedasticity robust and double clustered on industry (FF-12) and year. ***, **,* denote 1%, 5%, 10%
	significance, respectively.
		(1)	(2)	(3)	(4)
		IPO Size	IPO Size	SEO 1/0	SEO 1/0
	HTV	-0.019***	-0.011**	0.024***	0.042***
		(-6.21)	(-2.33)	(3.41)	(3.67)
	Leverage	0.041	0.030*	0.105**	0.121**
		(1.77)	(2.16)	(2.35)	(2.84)
	ln(Sales)	-0.019***	-0.022***	0.024***	0.016**
		(-4.72)	(-6.55)	(4.27)	(2.40)
	M/B	-0.054***	-0.054***	-0.006	-0.006
		(-8.57)	(-9.20)	(-0.39)	(-0.38)
	Cash/Assets	0.020	0.007	0.213	0.245
		(0.53)	(0.20)	(1.57)	(1.60)
	VCB	-0.005	-0.007	0.051	0.059*
		(-0.58)	(-0.85)	(1.81)	(2.01)
	ROA	0.053	0.064*	-0.233***	-0.233**
		(1.68)	(2.08)	(-3.45)	(-3.03)
	IPO Heat	-0.013	-0.014	0.006	0.005
		(-1.12)	(-1.31)	(0.17)	(0.16)
	Top Underwriter	0.030**	0.028**	-0.006	0.003
		(2.67)	(2.66)	(-0.24)	(0.10)
	UW Premium	-0.003	0.002	0.001	0.025
		(-0.07)	(0.05)	(0.01)	(0.31)
	IPO Size			-0.107	-0.102
				(-1.40)	(-1.28)
	First Day Return			0.305**	0.291**
				(2.81)	(2.84)
	Ln(Analyst)			0.106***	0.114***
				(4.01)	(4.23)
	1-Month Return			0.524***	0.528***
				(10.07)	(9.13)
	1-Month Std Dev			-0.815	-0.661
				(-1.56)	(-1.26)
	Constant	0.539***	0.565***	0.119	0.137
		(11.54)	(13.91)	(1.28)	(1.46)
	Observations	1,678	1,678	1,678	1,678
	R-squared	0.193	0.224	0.179	0.190
	Year FE	Y	Y	Y	Y
	Industry FE	N	Y	N	Y

Table 8 : HTV, peer group Q and peer group analysts

 8 Panel A of the table shows sortings of IPO firms into terciles based on peer group M/B and the number of peer group analysts according to the[START_REF] Hoberg | The Underwriter Persistence Problem[END_REF] Phillips (2010, 2016) classification. The table shows the value of the HTV proxy in the different terciles. Panel B reports regressions using IPO size as the dependent variable in odd columns and SEO likelihood in even columns. The main independent variable is the HTV proxy. Peer M/B is the average Q of the peer group firms according to the[START_REF] Hoberg | Product Market Synergies and Competition in Mergers and Acquisitions: A Text-Based Analysis[END_REF] 2016) classification. Peer Analyst is the peer group's number of analysts according to the same classification. Columns (3) and (4) use orthogonalized values of HTV, peer analyst and peer M/B. Models in odd columns include the controls of column (1) of Table3and even columns the controls of column (3) of Table3. All variables are defined as in appendix A1. All models include year fixed effects. Reported tstats in parentheses are heteroscedasticity robust and double clustered on industry (FF-12) and year. ***, **,* denote 1%, 5%, 10% significance, respectively.

	Panel A:	Low M/B	Mid		High M/B		Diff
	Low Analyst	-0.751	-0.570		-0.129	0.621***
	Mid	-0.388	0.272		0.713	1.101***
	High Analyst	0.203	0.864		0.873	0.670***
	Diff	0.954***	1.434***		1.002***	1.624***
	Panel B:	(1)	(2)	(3)	(4)	(5)	(6)
		IPO Size	SEO 1/0	IPO Size	SEO 1/0	IPO Size	SEO 1/0
	HTV	-0.007*	0.018*	-0.013***	0.016
		(-1.83)	(1.86)	(-3.30)	(1.58)
	Peer Analyst			0.008**	-0.004
				(2.94)	(-0.36)
	Peer M/B			0.002	0.014
				(0.47)	(1.01)
	HTV Orthogonalized					-0.017**	0.028*
						(-2.60)	(2.13)
	Peer Analyst Orthogonalized					-0.000	0.019
						(-0.00)	(1.55)
	Peer M/B Orthogonalized					0.002	0.019
						(0.29)	(0.85)
	Constant	0.435***	0.463***	0.383***	0.459***	0.421***	0.495***
		(6.27)	(4.67)	(5.19)	(3.55)	(6.04)	(4.84)
	Observations	2,265	2,265	2,265	2,265	2,265	2,265
	R-squared	0.253	0.141	0.244	0.133	0.242	0.134
	Controls	Y	Y	Y	Y	Y	Y
	Year FE	Y	Y	Y	Y	Y	Y
	Industry FE	N	N	N	N	N	N

Table 9 :

 9 Excluding VC backed IPOsColumns (1) and (2) of this table report regressions on IPO size measured as number of shares issued in the IPO scaled by the number of post-IPO shares. Columns (3) and (4) report regressions on an indicator variable taking value one if the firm conducts an SEO within 24 months following the IPO (SEO 1/0). Columns (5) and (6) report results from Cox hazard models with time from IPO to first SEO as the dependent variable. A positive coefficient estimate indicates a shorter time to failure (time to first SEO). We track the firms for two years following the IPO, i.e. our distributions are truncated at 730 days. We use the hard to value principal component (HTV) as our main explanatory variable. A three component PCA (HTV) including R&D scaled by sales, firm age and a high tech indictor. All variables are defined as in appendix 1. Reported t-stats in parentheses are heteroscedasticity robust and double clustered on industry (FF-12) and year. ***, **,* denote 1%, 5%, 10% significance, respectively.

		(1)	(2)	(3)	(4)	(5)	(6)
		IPO Size IPO Size SEO 1/0 SEO 1/0 Time to SEO Time to SEO
	HTV	-0.020*** -0.013*** 0.023** 0.037**	0.067*	0.120***
		(-6.70)	(-5.51)	(2.27)	(3.01)	(1.83)	(3.04)
	Leverage	0.054*** 0.047*** 0.092*	0.099*	0.441***	0.465***
		(3.67)	(3.33)	(1.82)	(2.04)	(3.05)	(3.48)
	ln(Sales)	-0.017*** -0.019*** 0.022**	0.016	0.091**	0.054
		(-4.22)	(-5.55)	(2.24)	(1.35)	(2.22)	(1.08)
	M/B	-0.048*** -0.048*** -0.004	-0.006	-0.002	-0.012
		(-8.26)	(-7.51)	(-0.37)	(-0.48)	(-0.04)	(-0.20)
	Cash/Assets	0.017	0.017	0.149	0.153	0.458	0.462
		(0.35)	(0.38)	(1.17)	(1.21)	(1.25)	(1.40)
	ROA	0.002	0.010	-0.135*	-0.117	-0.396*	-0.269
		(0.11)	(0.54)	(-2.03)	(-1.63)	(-1.77)	(-1.09)
	IPO Heat	-0.003	-0.005	-0.015	-0.013	-0.156	-0.126
		(-0.28)	(-0.45)	(-0.43)	(-0.38)	(-1.62)	(-1.29)
	Top Underwriter	0.022**	0.020**	0.004	0.008	0.049	0.094
		(2.24)	(2.34)	(0.28)	(0.45)	(0.80)	(1.30)
	UWAgencytotal	0.041	0.042	-0.051	-0.038	-0.164	-0.088
		(1.15)	(1.14)	(-0.70)	(-0.52)	(-0.53)	(-0.30)
	IPO Size			-0.092	-0.101	-0.387	-0.475
				(-1.49)	(-1.59)	(-1.25)	(-1.42)
	First Day Return			0.085** 0.093**	0.398***	0.450***
				(2.66)	(2.91)	(2.76)	(2.96)
	Ln(Analyst)			0.225** 0.221**	1.106***	1.158***
				(2.72)	(2.70)	(2.95)	(2.88)
	1-Month Return			0.559*** 0.565*** 2.427***	2.516***
				(6.92)	(7.23)	(9.24)	(8.08)
	1-Month Std Dev			-0.383	-0.271	-2.592	-2.109
				(-0.75)	(-0.46)	(-0.75)	(-0.62)
	Constant	0.504*** 0.522*** 0.167*	0.187*
		(12.54)	(12.92)	(1.94)	(2.13)
	Observations	1,883	1,883	1,883	1,883	1,883	1,883
	R-squared	0.189	0.205	0.155	0.163
	Year FE	Y	Y	Y	Y	Y	Y
	Industry FE	N	Y	N	Y	N	Y

Table 11 :

 11 Monthly ReturnsThis table shows regressions on SEO likelihood (SEO 1/0). The main independent variable is the HTV proxy and the monthly returns following the IPO. All models include the controls of column (3) in Table.All variables are defined as in appendix A1. Reported t-stats in parentheses are heteroscedasticity robust and double clustered on industry (FF-12) and year. ***, **,* denotes 1%, 5%, 10% significance, respectively.

	(1)	(2)
	SEO 1/0	SEO 1/0

Table 12 :

 12 Investors' risk aversionThis table shows regressions on IPO size in columns (1) and (2) and SEO likelihood (SEO 1/0) in columns (3) and (4). The main independent variables are indicators taking the value of 1 if the firm/month belong to the the top tercile of the HTV proxy, bottom tercile of[START_REF] Baker | Investor Sentiment and the Cross-Section of Stock Returns[END_REF] sentiment index, top tercile of[START_REF] Zhou | Variance Risk Premia, Asset Predictability Puzzles, and Macroeconomic uncertainty[END_REF] variance risk premia, and their intercations. Models (1) and (2) include the controls of column (1) of Table3 and model (3) and (4) the controls of column (3) of Table3. All variables are defined as in appendix A1. Reported t-stats in parentheses are heteroscedasticity robust and double clustered on industry (FF-12) and year. ***, **,* denotes 1%, 5%, 10% significance, respectively.

	(1)	(2)	(3)	(4)
	IPO Size	IPO Size	SEO 1/0	SEO 1/1

Table 13 :

 13 Alternative explanationsThis table shows regressions on IPO size in columns (1) and (

  Table3 and model (3) the controls of column (3) of Table3. All variables are defined as in appendix A1. Reported t-stats in parentheses are heteroscedasticity robust and double clustered on industry (FF-12) and year. ***, **,* denotes 1%, 5%, 10% significance, respectively.

		(1)	(2)	(3)
		IPO Size	IPO Size	SEO 1/0
	HTV	-0.014***	-0.014**	0.020*
		(-4.37)	(-3.02)	(1.91)
	Negative Cash flow		0.013	0.003
			(0.77)	(0.15)
	Net Cash Flows		0.128***	0.066
			(3.32)	(0.42)
	Negative Cash Flows x Net Cash Flows		-0.086*	-0.133
			(-2.12)	(-0.81)
	First Day Return	0.119***		0.241***
		(6.87)		(4.95)
	Constant	0.451***	0.395***	0.383***
		(12.71)	(5.94)	(4.17)
	Observations	3,517	2,802	2,802
	R-squared	0.251	0.240	0.124
	Controls	Y	Y	Y
	Year FE	Y	Y	Y
	Industry FE	N	N	N

Table 14 :

 14 IPO process information revelationThe table below reports OLS models where IPO size is the dependent variable in odd columns and SEO likelihood in even columns. The main independent variable is the HTV proxy. Models in odd columns include the controls of column (1) of Table3and even columns the controls of column (3) of Table3. All variables are defined as in appendix A1. Reported t-stats in parentheses are heteroscedasticity robust and double clustered on industry (FF-12) and year. ***, **,* denotes 1%, 5%, 10% significance, respectively.

		(1)	(2)	(3)	(4)
		IPO Size	SEO 1/0	IPO Size	SEO 1/0
	HTV	-0.022***	0.036**	-0.019***	0.055***
		(-6.50)	(3.08)	(-3.65)	(3.61)
	Abs(Revision)	-0.057	0.037	-0.057	0.045
		(-1.68)	(0.36)	(-1.77)	(0.46)
	IPO cost	-0.027***	-0.018	-0.025***	-0.027
		(-4.07)	(-1.12)	(-5.19)	(-1.56)
	Ln(Analyst)	-0.001	0.075**	0.005	0.085**
		(-0.12)	(2.54)	(0.54)	(3.09)
	First Day Return	0.088***	0.292**	0.086***	0.291**
		(3.90)	(2.90)	(3.74)	(3.11)
	Constant	0.739***	0.502*	0.734***	0.582**
		(11.19)	(2.21)	(16.82)	(2.34)
	Observations	1,927	1,927	1,926	1,926
	R-squared	0.242	0.115	0.256	0.130
	Controls	Y	Y	Y	Y
	Year FE	Y	Y	Y	Y
	Industry FE	Y	Y	Y	Y

As in prior studies such as[START_REF] Baker | Market Timing and Capital Structure[END_REF] andAlti (2006), we require availability of pre-IPO accounting data. This constrains our sample size, which is similar to that in the mentioned prior studies.

The analysis of the effects of B, θL and θH on the desirability of a staged equity offering is more complex as these parameters shift 𝑝̅ and 𝑝 into the same direction. However, it can be shown that the public listing becomes more attractive the larger is the ratio ∆H/∆L, since the expected market value of the firm is increasing in ∆H/∆L. The attractiveness of a staged equity offering compared to a single, large IPO is inversely related to ∆H/∆L since the smaller is ∆H/∆L the more valuable is experimentation via a small IPO.

Focusing on price revisions that occur during the first trading day, our model would also provide an explanation for IPO underpricing (overpricing) without a need to resort to asymmetric information, behavioral biases or other even more complicated explanations (see, e.g.,Ljungqvist, 2007, for a survey). In our empirical application, we use a price revision period of one moth.

Our sample size is smaller compared to[START_REF] Hertzel | Public Market Staging: The Timing of Capital Infusions in Newly Public Firms[END_REF] due to our restriction that firms need to have data available in COMPUSTAT for the year-end prior to the IPO. Also, their sample is partially hand-collected from 10K filings, whereas we rely on the SDC Platinum database.

The one month time period coincides with our return measurement period and thereby we avoid look-ahead bias in the returns.

http://bear.warrington.ufl.edu/ritter

The number of observations deviate between the three terciles due to equal values of the sorting variable.

However, consistent withAngrist and Pischke (2009), we obtain similar estimates in untabulated tests using Probit.

We do not include this set of additional controls in Columns (1) and (2) of Table3, as they are determined post-IPO. However, in our robustness tests we consider, First day return (Table12), analyst following (Table

13) and volatility (Table8) in IPO size regressions.

Our results are in line with[START_REF] Brau | Do Secondary Shares in the IPO Process have a Negative Effect on Aftermarket Performance?[END_REF], who report no-relation between insider participation and post-IPO performance.

Table 3: Base case regression models

Columns (1) and (2) of this table report regressions on IPO size measured as number of shares issued in the IPO scaled by the number of post-IPO shares. Columns (3) and (4) report regressions on an indicator variable taking value one if the firm conducts an SEO within 24 months following the IPO (SEO 1/0). Columns ( 5) and ( 6) report results from Cox hazard models with time from IPO to first SEO as the dependent variable. A positive coefficient estimate indicates a shorter time to failure (time to first SEO). We track the firms for two years following the IPO, i.e. our distributions are truncated at 730 days. We use the hard to value principal component (HTV) as our main explanatory variable. A three component PCA (HTV) including R&D scaled by sales, firm age and a high tech indictor. All variables are defined as in appendix 1. Reported t-stats in parentheses are heteroscedasticity robust and double clustered on industry (FF-12) and year. ***, **,* denote 1%, 5%, 10% significance, respectively.

( (