Othmane-Latif Ouabi
email: oouabi@georgiatech-metz.fr

Jiawei Yi

Neil Zeghidour
email: neilz@google.com

Nico F Declercq
email: declercq@gatech.edu

Matthieu Geist
email: mfgeist@google.com

Cédric Pradalier
email: cedricp@georgiatech-metz.fr

Polygonal Shapes Reconstruction from Acoustic Echoes Using a Mobile Sensor and Beamforming

Keywords: mapping, acoustic echoes, beamforming

Mapping a structure, such as a metal plate in a ship hull, using acoustic echoes typically requires making assumptions on its shape (e.g. rectangular). This work introduces a more general beamforming approach to recover the geometry of arbitrary polygonal-shaped bounded areas from acoustic reflections. Our method only requires a single omnidirectional emitterreceiver acoustic device mounted on a mobile platform. We apply beamforming to the acoustic measurements in the geometrical boundary space. We subsequently retrieve the edges from the beamforming results via the minimization of a regularized cost criterion, using a simulated annealing optimizer. We also design a boundary rejection criterion so that their exact number can be recovered based only on a specified upper bound. We assess our method on different geometries in a simulation environment and a real-world setting using ultrasonic guided waves measurements. The results demonstrate that it is efficient for achieving the targeted objectives.

I. INTRODUCTION

Acoustic mapping from echoes is a significant challenge that can enable a wide variety of applications, in particular in robotics. When an acoustic wave is emitted by an embedded transmitter, it can propagate in all directions around the emitter. It can be reflected by environmental features (such as the walls of an enclosed space, for instance). Simultaneously, co-located with the emitter, a receiver can receive the acoustic echoes. With such a setup, one objective is the localization of the acoustic reflectors using the measurements acquired along the robot trajectory. For instance, room shape reconstruction in an indoor environment can be made possible using a single mobile unit [START_REF] Saqib | A model-based approach to acoustic reflector localization with a robotic platform[END_REF]. More generally, acoustic Simultaneous Localization and Mapping (SLAM) [START_REF] Kreković | EchoSLAM: Simultaneous localization and mapping with acoustic echoes[END_REF], [START_REF] Krekovic | Omnidirectional bats, point-to-plane distances, and the price of uniqueness[END_REF] can be achieved to additionally recover the acquisition positions. Another use case is localization and mapping based on Ultrasonic Guided Waves (UWGs) on a large structure such as a ship hull, where the ultrasonic echoes originate from reflections, on the individual plate boundaries, of the wave propagating in the material [START_REF] Ouabi | A FasSLAM Approach Integrating Beamforming Maps for Ultrasound-based Robotic Inspection of Metal Structures[END_REF],

This work is part of the BugWright2 project. This project is supported by the European Commission under grant agreement 871260 -BugWright2. [START_REF] Pradalier | Onplate localization and mapping for an inspection robot using ultrasonic guided waves: A proof of concept[END_REF]. These examples are all instances of the "structure from sound (and motion)" problem.

While acoustic localization and mapping using sensor arrays receive sustained attention in the literature [START_REF] Salvati | Sound Source and Microphone Localization from Acoustic Impulse Responses[END_REF], [START_REF] Michaud | 3D Localization of a Sound Source Using Mobile Microphone Arrays Referenced by SLAM[END_REF], the use of a single omnidirectional sensor remains a challenging task for practical implementation. Indeed, as the measurements contain several first-order as well as high-order echoes, acoustic localization and mapping necessitate echo-detection, identification and association methodologies that are challenging to establish for reverberant and noisy environments [START_REF] Worley | Acoustic Echo-Localization for Pipe Inspection Robots[END_REF].

Acoustic echoes in the air have been used to recover a room shape from wall reflections [START_REF] Peng | Room shape reconstruction with a single mobile acoustic sensor[END_REF], or to achieve acoustic SLAM [START_REF] Kreković | EchoSLAM: Simultaneous localization and mapping with acoustic echoes[END_REF], [START_REF] Krekovic | Omnidirectional bats, point-to-plane distances, and the price of uniqueness[END_REF]. However, these works are only tested in simulation. They rely on essential assumptions, such as only first-order echoes are being detected, the echo-wall association is being flawlessly known, or the number of walls to recover is being known a priori. Due to noisy measurements, these assumptions may not always hold for a real-world scenario.

Recent works in the literature focus on acoustic SLAM using Ultrasonic Guided Waves (UGWs) that can propagate over large distances inside materials. In [START_REF] Pradalier | Onplate localization and mapping for an inspection robot using ultrasonic guided waves: A proof of concept[END_REF], an online approach is designed to simultaneously recover the geometry of individual metal plates while localizing a mobile sensor, but it lacks robustness w.r.t. to detection errors. A second work [START_REF] Ouabi | A FasSLAM Approach Integrating Beamforming Maps for Ultrasound-based Robotic Inspection of Metal Structures[END_REF] achieves acoustic SLAM by relying on beamforming [START_REF] Van Veen | Beamforming: A versatile approach to spatial filtering[END_REF]. This approach shows promising results, as it does not necessitate explicit echo retrieval and association prone to errors. It has been subsequently improved in [START_REF] Ouabi | Combined Grid and Feature-based Mapping of Metal Structures with Ultrasonic Guided Waves[END_REF] through the combination of beamforming and a probabilistic grid representation that builds on the estimation of the first acoustic echo to limit beamforming errors due to interference. However, the beamforming approach for mapping the plate boundaries is limited to rectangular geometries.

The objective of this work is to generalize the approach presented in [START_REF] Ouabi | A FasSLAM Approach Integrating Beamforming Maps for Ultrasound-based Robotic Inspection of Metal Structures[END_REF] and [START_REF] Ouabi | Combined Grid and Feature-based Mapping of Metal Structures with Ultrasonic Guided Waves[END_REF] to recover arbitrary 2-dimensional polygonal geometries from acoustic echoes. It relies on beamforming in the geometrical boundary space and the single detection of the closest boundary to map the inner surface of the geometric shape. The geometry estimate is then constructed via the minimization of a regularized cost function using a simulated annealing optimizer to select boundaries, on the beamforming results, consistent with the mapped inner space. We also design a boundary rejection criterion to recover their exact number based only on a specified upper bound. We test our approach in simulation for various geometries and actual conditions using experimental ultrasonic measurements acquired on a metal plate sample. The results show that it can successfully recover the geometric shape of the reverberant environment, the exact number of boundaries, and has the same performance as the baseline method in [START_REF] Ouabi | A FasSLAM Approach Integrating Beamforming Maps for Ultrasound-based Robotic Inspection of Metal Structures[END_REF] for recovering rectangular shapes.

II. METHOD

We consider a setup where a single (collocated) emitter/receiver acoustic device is driven to acquire acoustic measurements z 1..T (t) at positions x 1..T in a delimited space, where x i = [x i , y i ] T contains the two-dimensional position. At the i-th scanning position, a pulse s(t) is used to emit an omnidirectional acoustic wave. The received signal z i (t) contains the reflections of the incident wave on the environment boundaries. We assume that noise-free odometry information (i.e linear and rotational displacement between the positions of the successive measurements) is available. For a robotic application, it is typically provided by the wheels encoders, but other systems such as an Inertial Measurement Unit (IMU) can also be integrated for better accuracy. We also assume that, at each position x i , a measurement of the range ρ i to the closest boundary is available by processing the acoustic data. This measurement may only be subject to little noise in practice, as it may be one of the most energetic echoes. Based on all range measurements ρ 1:T , the surface A T is defined as the unions of all disks centered at sensor positions x i , and with the radius equal to the detected range ρ i . This surface accounts for areas where a boundary cannot be present. The objective is to estimate the shape of the environment M that we assume to be a 2D convex polygon using measurements and odometry. It should be noted that only the mapping problem is addressed here. Still, the proposed approach is expected to integrate seamlessly within the SLAM framework in [START_REF] Ouabi | A FasSLAM Approach Integrating Beamforming Maps for Ultrasound-based Robotic Inspection of Metal Structures[END_REF] to account for odometry noise, and the method in [START_REF] Ouabi | Combined Grid and Feature-based Mapping of Metal Structures with Ultrasonic Guided Waves[END_REF] to consider detection errors on the range measurements ρ i .

A. Measurement model

The acoustic measurements contain successive reflections of the excitation on the boundaries. Under the assumption that the propagation is linear, and the medium is homogeneous and isotropic, we can use the image source model [START_REF] Kuttruff | Room acoustics[END_REF]:

z i (t) = pi,j ∈I(xi,M) g(||p i,j -x i ||, t) * s(t) + n i (t), ( 1
)
where g is the acoustic transfer function of the isotropic medium, I(x i , M) is the set of image source positions for a real source in x i , and n i is additive Gaussian noise that we assume to be spatially and temporally white.

B. Delay and sum beamforming

The geometry M to estimate is assumed to be a 2D convex polygon that can be represented by its linear boundaries: M = {r l , θ l } l=1..L where the parameters (r l , θ l ) define the line equation in the 2D plane with:

x • cos θ l + y • sin θ l -r l = 0, (2)
expressed in a fixed frame which origin will be chosen as the initial robot position, and the x axis will be aligned with the initial robot heading. The number L of boundaries is assumed a priori unknown. The adopted notations are precised in Fig. 1.

To estimate the map M, we rely on beamforming as previous works [START_REF] Ouabi | A FasSLAM Approach Integrating Beamforming Maps for Ultrasound-based Robotic Inspection of Metal Structures[END_REF]. The approach is recalled here for completeness. We first evaluate the correlation measurements:

zi (r) = ⟨z i (t), ẑ(r, t)⟩ ⟨z i (t), z i (t)⟩⟨ẑ(r, t), ẑ(r, t)⟩ , (3)
where ẑ(r, t) = ĝ(2r, t) * s(t) is the expected received signal for a boundary situated at a distance r from the receiver, and ĝ is a model of the transfer function. We then retrieve the envelope: e i (r) = |z i (r) + jH(z i )(r)|, where H denotes the Hilbert transform operator. Hence, the positions of the local maxima of e i yield the distances to the boundaries, and ρ i can be selected as the earliest one. Next, we assess the likelihood of existence for any line boundary (r, θ) by constructing the beamforming map based on the correlation measurements and sensor positions:

L T (r, θ) = T i=1 e i (d i (r, θ)) , (4)
where d i (r, θ) = |x i • cos θ + y i • sin θ -r| is the distance between the receiver and the considered line. It is to be noted that only first-order reflections are considered in the above equation, as we reason on individual lines.

C. Spatially-regularized cost function

The previous mapping approach in [START_REF] Ouabi | A FasSLAM Approach Integrating Beamforming Maps for Ultrasound-based Robotic Inspection of Metal Structures[END_REF] can only be used to recover rectangular shapes by retrieving exactly four local maxima of L(r, θ) through the maximization of the cumulative likelihood. We propose to extend this approach to recover, in principle, arbitrary 2D polygonal shapes. The estimated geometry has the form: M = {r k , θk } k=1..K , where the maximum number of boundaries K is such that K ≥ L. This is a very minimal hypothesis, as K can be set much higher than the unknown L.

Next, for a given robot trajectory x 1:T , we determine the inner surface A T based on measuremments ρ 1:T , and the beamforming map L T calculated with Eq. ( 4) that we normalize by its maximum value. We then consider the cost:

c( M) = -   (r k ,θ k )∈ M L(r k , θ k ) + λ M A T M A T   , (5)
where the first term aggregates the likelihood of presence of the estimated boundaries, and the second term is the well known Intersection over Union (IoU), widely used for computer vision purposes [START_REF] Rahman | Optimizing intersection-overunion in deep neural networks for image segmentation[END_REF]. In Eq. ( 5) the surfaces M and A T are as described in Fig. 1. The IoU term (that we will simply write as IoU( M, A T )) is always between 0 and 1. It is zero when M and A T have an empty intersection, and it is 1 when the two surfaces are equal. λ is a positive parameter that balances the importance between the boundaries likelihood and IoU terms. Our geometry estimate is then defined as:

MT = arg min M c( M). (6)
Hence, the minimization of the cost criterion leads to maximizing the cumulative likelihood of the estimated boundaries and maintains consistency between the estimated inner surface of M with that of A T . In other words, the IoU in Eq. ( 5) acts as a regularization term for selecting likely boundaries on the beamforming results.

D. Shape estimation with simulated annealing

An optimization strategy is required to recover a geometry estimate by minimising the cost. Simulated annealing (SA) [START_REF] Kirkpatrick | Optimization by simulated annealing[END_REF] is a relatively simple and efficient approach for estimating a global minimum of a non-linear cost function. This metaheuristic is inspired by metallurgy, where, to form a perfect crystal (which corresponds to the state of minimal energy), a pure liquid substance is slowly cooled. A random perturbation is applied to the current state at each iteration of the optimization process. The perturbation is systematically accepted if the energy is decreased. It is accepted according to a Boltzmann probability distribution when the energy is increased to escape local minima. The temperature, which is a hyperparameter that controls the acceptance rate, is decreased slightly after each iteration, so that the probability of increasing the energy goes to zero. Even though simulated annealing offers no guarantees that the optimum will be found, this strategy can lead to satisfying solutions in a relatively restricted number of iterations. In the following, we detail the different steps of our custom implementation for mapping arbitrary polygonal geometries from acoustic echoes.

1) Initialization: First, the geometry estimate is randomly initialized with MT = {r k , θk } k=1..K , where the rk and θk are sampled from independent uniform distributions.

2) State disturbance: At iteration p of the optimization process, each boundary of the estimate is iteratively disturbed with a random process. For the k-th boundary, we sample:

rk = max(r k + ∆ r • χ 3 r , 0) θk = θk + ∆θ • χ 3 θ mod 2π , (7)
where ∆ r and ∆ θ are positive parameters, and χ r and χ θ are independently sampled from uniform distributions: U([-1, 1]).

They are raised to the power 3 so that small variations are more likely. Still, large variations (i.e in the order of ∆ r and ∆ θ ) can be possible to explore regions of the search space that are far occasionally from the current state. Furthermore, to speed up the optimizer convergence, we determine:

(r k , θk ) ← arg max (r,θ)∈V (r k , θk ) L T (r, θ), (8)
where V (r k , θk ) denotes the set of boundaries that lie in the neighborhood of (r k , θk ) that is to be appropriately defined. The purpose of this step is to only select (r k , θk ) that are local maxima, which is a property that is expected for the true boundaries. The former edge (r k , θk ) in MT is then substituted to the sampled one, yielding a new geometry ∼ M T . 3) Simulated annealing acceptance process: The new candidate map is kept as the current state following an SA acceptance process. First, we determine the cost variation ∆c = c(

∼ M T ) -c( MT ). Let's χ be a sample from U([0, 1]).
The candidate geometry is kept only if we have:

χ < exp - ∆c T (p) , (9)
where T (p) is the temperature at iteration p of the optimizer. Hence, if the cost is decreased, the new map is systematically kept as the new current state MT . Otherwise, it is kept according to the Boltzmann distribution to escape local minima.

4) Edge filtering:

As it is expected that the edges number parameter K is overestimated, some of them may not contribute significantly to the definition of the recovered geometry once the optimizer has converged. To derive a proper geometry with a correct number of boundaries, we apply a rejection criterion to remove non-informative edges. We iterate over each boundary (r k , θk ) in MT , and we remove it to yield a new candidate geometry ∼ M T . We then assess whether:

IoU( ∼ M T , A T ) -IoU( MT , A T ) IoU( MT , A T ) < ϵ, (10)
where we will take ϵ = 10 -2 . If the IoU variation is lower than the threshold, then the k-th boundary can be removed, as its contribution to the definition of the geometric surface is considered negligible. Then, the geometry ∼ M T can be taken as the new candidate MT , and the process is iterated on the next boundary. Also, the rejection criterion is not applied if the candidate map MT has 3 edges (which is a lower bound for a polygonal shape). III. EXPERIMENTS We perform different experiments to assess the efficiency of our method for recovering arbitrary 2-dimensional polygonal geometries from acoustic echoes acquired over a mobile unit trajectory. We perform simulation tests on three shapes and validation tests using experimental acoustic data acquired on a rectangular metal plate. For this latter scenario, the baseline method in [START_REF] Ouabi | A FasSLAM Approach Integrating Beamforming Maps for Ultrasound-based Robotic Inspection of Metal Structures[END_REF] is also applied for comparison.

A. Simulation results

We consider three simulated scenarios, where the geometry to be recovered has either 3, 4 or 5 boundaries. The robot paths are randomly generated with successive linear and rotational displacements. The ultrasonic measurements acquired over the path are simulated using the Image source model and the ultrasonic model g used in [START_REF] Ouabi | A FasSLAM Approach Integrating Beamforming Maps for Ultrasound-based Robotic Inspection of Metal Structures[END_REF]. It should be noted that such a data model leads to over-energetic high-order reflections, as acoustic absorption from the boundaries is not taken into account. The excitation s(t) is a two tone bursts sinusoidal wave at 100 kHz. We add Gaussian noise on the measurements to maintain a fixed signal-to-noise ratio SNR = 0. The considered geometries and paths are provided in Fig. 2.

We run 10 repetitions of the optimizer for each scenario to assess the repeatability of our approach. The optimizer is set with parameters λ = 3, T (p) = 5/p 0.8 , and V (r, θ) is chosen as the set of edges with line coordinates within the rectangular area centered in (r, θ), and with size 5.4cm × 48 • . Also, we use K = 10 as the maximum number of edges. Fig. 3 shows the evolution of the cost, as well as the number of estimated edges after use of the edge rejection criterion during 300 optimizer iterations and for the 10 repetitions. We also display the evolution of IoU( M, M), which is the IoU between the estimated inner surface (based on the estimated edges) and the ground truth surface M. Hence, an IoU of 1 translates into a perfect geometry estimation. The results demonstrate that, for all the considered scenarios, our approach can seamlessly recover, after convergence, both the actual number of boundaries and the geometric shape.

B. Experimental results

To validate our approach, we run it using experimental ultrasonic data acquired on a rectangular aluminium metal plate by moving, by hand, two nearly collocated contact piezoelectric transducers for emission and reception. More details on the experimental setup and acquisition procedure are provided in [START_REF] Ouabi | A FasSLAM Approach Integrating Beamforming Maps for Ultrasound-based Robotic Inspection of Metal Structures[END_REF]. The optimizer is set with the exact same parameters than those used for the simulation tests to highlight the robustness of our approach. Also, we do not use ground- truth ranges ρ 1:T , but apply a standard peak detection method on the correlation signals to estimate them. This only leads to a slight estimation error due to high SNR. The considered simulated robot trajectory is depicted in Fig. 4a. Fig. 4 shows the results and the evolution of the performance criteria for 10 repetitions of the optimization process. For comparison, we also display the IoU achieved with the baseline method presented in [START_REF] Ouabi | A FasSLAM Approach Integrating Beamforming Maps for Ultrasound-based Robotic Inspection of Metal Structures[END_REF], which does not require iterative optimization, but is restricted to rectangular shapes. We can see that the estimated number of edges converges seamlessly to 4, while the IoU goes to 1 for all the repetitions. Thus, our method achieves the performance of the baseline approach, despite not being provided with the ground-truth number of edges nor the rectangular constraint. Overall, the results demonstrate the potential of our approach for recovering polygonal shapes from acoustic echoes using a single mobile omnidirectional emitter/receiver device and beamforming.

IV. CONCLUSION

We have developed a new mapping approach to recover the geometry of unknown 2D polygonal structures from acoustic echos using a single omnidirectional mobile sensor. It combines delay-and-sum beamforming in the geometric boundary space and a simulated annealing optimizer to retrieve boundary estimates from the beamforming results. Contrary to previous works, our approach does not rely on solid hypotheses on echo detection and identification and can recover, in principle, arbitrary polygonal shapes.

The approach is tested in a simulation environment. The results are promising, with a correct and efficient reconstruction of various geometries, without prior knowledge of the number of boundaries to recover. The approach is also validated on experimental acoustic data acquired in a reallife setup to demonstrate its robustness. The results show that the approach achieves the same performance as the baseline method restricted to rectangular geometries. The optimization efficiency shall be improved, in future works, so that the method can be run online. Furthermore, the method shall be integrated within a SLAM framework, as the sensor positions are to be estimated in practice. Its extension to recover 3D volumes shall also be investigated.

Fig. 1 :

 1 Fig. 1: Schematic of the problem setup and notations. An example of shape to estimate is represented by the red polygon. A candidate geometry estimate is represented by the set of green lines, which (r, θ) coordinates are only provided for one of them. The area A formed with a single range measurement ρ is represented by the lighter area. The inner surfaces of A and M are delimited with the discontinuous dashed lines.

Fig. 2 :

 2 Fig. 2: The different simulated setups along with an instance of mapping results. On the left of each sub-figure, the outline of the ground-truth geometry is provided in discontinuous red, the green lines account for the estimated edges, and the robot path and acquisition positions are represented in black. The corresponding beamforming maps are provided on the right, where the red rectangles indicate the retrieved edges. The edge coordinates are expressed in the fixed frame, which origin is the initial sensor position O, and the x-axis is aligned with the initial robot heading.

Fig. 3 :

 3 Fig. 3: Evolution of the performance criteria for the 3 scenarios, and for 10 repetitions of the optimizer. The top plots show, in blue, the evolution of the costs. The middle plots depict the estimated number of edges after application of the edge rejection criterion. The bottom plots show the IoU between estimated and ground-truth surfaces. The median values are in red.

  (a) Mapping setup and results for the experimental scenario. (b) Evolution of performance criteria during the optimization.

Fig. 4 :

 4 Fig. 4: Results for the experimental setup. (a, left) shows the simulated sensor path (in black), the plate outline (red) and estimated edges (green). The corresponding beamforming map is provided on the right, where the red rectangles indicate the retrieved edges. (b) shows the evolution of the loss, the number of estimated edges and IoU for 10 repetitions of the optimizer. The black horizontal line on the IoU plot is the value achieved with the baseline method [1]. The median values are in red.


Hashem Dezhbakhsh

Daniel Levy
email: daniel.levy@biu.ac.il

Interpolation and shock persistence of prewar U.S. macroeconomic time series: A reconsideration*

Keywords: JEL Classification: C01, C02, E01, E30, N10 Linear Interpolation, Random Walk, Shock-Persistence, Nonstationary series, Periodic nonstationarity, Stationary series, Prewar US Time Series

The U.S. prewar output series exhibit smaller shock-persistence than postwar-series. Some studies suggest this may be due to linear interpolation used to generate missing prewar data. Monte Carlo simulations that support this view generate large standard-errors, making such inference imprecise. We assess analytically the effect of linear interpolation on a nonstationary process. We find that interpolation indeed reduces shock-persistence, but the interpolated series can still exhibit greater shock-persistence than a pure random walk. Moreover, linear interpolation makes the series periodically nonstationary, with parameters of the data generating process and the length of the interpolation time-segments affecting shock-persistence in conflicting ways.

1

Introduction

Most prewar U.S. data including output, CPI, etc. were observed only in benchmark years, several years apart. The missing observations were usually reconstructed by linear interpolation of the benchmark observations, sometimes padded with a serially correlated term (See, e.g., [START_REF] Friedman | Monetary Trends in the US and the UK[END_REF], Romer 1989[START_REF] Balke | The estimation of prewar gross national product: methodology and new evidence[END_REF][START_REF] Johnston | The annual real and nominal GDP for the United States, 1790-1928[END_REF]. Interpolation was also used to construct many historical series for the UK (Measuring Worth 2022). 1Studies find that US prewar output data is less shock-persistent than the U.S. postwar output data. 2US prewar-data also has different shock-persistence properties than other countries' prewar data [START_REF] Cogley | International evidence on the size of the random walk in output[END_REF]. Stock and Watson (1986) suggest that this difference could be due to linear interpolation of missing values for the prewar data. Although, as Romer (1989) notes, much of our knowledge of the macroeconomy during the prewar periods is based on these interpolated data, the effect of interpolation on shock-persistence has been rarely examined. 3 An exception is [START_REF] Jaeger | Shock persistence and the measurement of prewar output series[END_REF] who assesses the effect of linear interpolation using Monte Carlo simulations for a random-walk with MA(1) errors. He generates a 50-observation time series, "original series," and a corresponding "interpolated series," where the 10 th , 20 th , 30 th , 40 th and 50 th observations are generated by the above data generating process (DGP), and the rest are linearly interpolated. He then calculates the variance-ratio measure of shockpersistence for the "original" and "interpolated series" over 500-replications, where variance-ratio is defined as the ratio of the variance of the series' s-period growth (long variance) to the variance of the series' 1-period growth (short variance), and finds that linear interpolation indeed reduces shockpersistence. While Jaeger's experiments are well-designed, his estimates are very imprecise. For example, 2-SE confidence-bounds around his variance-ratio estimates fall within (0, 2+) or even (0, 3+), making it impossible to infer whether or not the ratio is less than 1 (stationary) or larger than 1 (non-stationary).

Thus, one cannot infer from these results that interpolation necessarily reduces shock-persistence.

We examine analytically the effect of interpolation on the shock-persistence of a non-stationary series. The advantage of analytical approach is that it can identify interpolation effects that are distinct and not diluted by sampling-variation, which is the hallmark of simulations. We assume the same DGP as [START_REF] Jaeger | Shock persistence and the measurement of prewar output series[END_REF]-a random-walk with MA(1) errors, for parsimony and comparability. We first derive the variance-ratio of the original, non-interpolated series, and then develop a linear interpolation model for the series with a general benchmark cycle, replacing all non-benchmark data points with their linearly interpolated values and a moving average padding. We then derive the variance-ratio for the interpolated series and compare it to the variance ratio of the original series to ascertain the effect of interpolation on shock-persistence.

Our results provide analytical support for [START_REF] Jaeger | Shock persistence and the measurement of prewar output series[END_REF] Monte Carlo findings. Additionally, we uncover a few other interpolation-caused effects. Although interpolation reduces shock-persistence, interpolated series may still exhibit high shock-persistence with variance-ratios greater than 1. Also, linear interpolation makes the series periodically non-stationary, with parameters of the DGP and the length of the interpolated time segments affecting shock-persistence in conflicting ways.

We proceed as follows. In section 2, we present the DGP and derive its shock-persistence measure analytically. In section 3, we develop the interpolation model for this DGP and analytically derive the corresponding shock-persistence measures. In section 4, we compare the results for non-interpolated and interpolated series. Section 5 concludes.

Shock-persistence of a random-walk with MA(1) errors

Following [START_REF] Jaeger | Shock persistence and the measurement of prewar output series[END_REF], we assume a random-walk with MA(1) errors and no-drift: As [START_REF] Jaeger | Shock persistence and the measurement of prewar output series[END_REF] notes, this setup captures the main features of the interpolation procedure, as described by Romer (1986Romer ( , 1989)). The model is also parsimonious: it is simple, yet it can capture the dynamics of many macroeconomic time series, and thus it and similar models are frequently employed in macroeconomic time series analysis (e.g., [START_REF] Nelson | Trends and random walks in macroeconomic time series[END_REF][START_REF] Campbell | Are output fluctuations transitory?[END_REF][START_REF] Cochrane | How big is the random walk in GNP?[END_REF], Cogley 1990, etc.).

To measure shock-persistence, we follow [START_REF] Jaeger | Shock persistence and the measurement of prewar output series[END_REF] and others [START_REF] Cochrane | How big is the random walk in GNP?[END_REF][START_REF] Cogley | International evidence on the size of the random walk in output[END_REF][START_REF] Leung | Changes in the behavior of output in the United Kingdom, 1856-1990[END_REF], Levy and Dezhbakhsh 2003, etc.) by using variance-ratio measure which is defined as

2 2 , 1, Y s Y Y V s σ σ = , where
( ) 2 , var s Y T T s Y Y σ - = -
is the variance of the series' s-period growth-"longvariance," and

( ) 2 1, 1 var Y T T Y Y σ - = -
is the variance of the series' 1-period growth-"short-variance."

The short variance for the above process is

( ) ( ) 2 1, 1 0 var var Y T T T Y Y U σ γ - = - = = where ( ) 2 2 0 1 ε γ θ σ = + .
The long variance for this process is given by

( ) ( ) ( ) 2 , 1 0 1 2 2 1 0 0 var var var 2 cov , s Y T T s s T j j s s s T j T j T i j j i j Y Y U U U U σ - - - = - - - - - -- = = = = -   =     = + ∑ ∑ ∑∑ ( ) 0 1 2 3 1 0 1 2 2 2 2( 1) 2( 2) 2( 3) 2 2( 1) 1 2( 1) s s s s s s s s s ε ε γ γ γ γ γ γ γ θ σ θσ - = + - + - + - + + = + - = + - -  where 0 γ is given above, 2 1 ε γ θσ = -
, and 0

j γ = for 2 j ≥ .
The variance-ratio for the original, non-interpolated random walk with MA(1) series, is therefore

( ) ( ) 2 , 2 1, 2 2 2 2 2 1 2( 1) 1 s Y Y Y V s s s s ε ε ε σ σ θ σ θσ θ σ = + - - = + ( ) 2 2(1 ) 1 1 s s θ θ - = + + (1)
Y y = , 2 0,2 Y y = , …, 0, s s Y y = , 1 1,1 s Y y + = , …, , st i t i Y y + = .
In this notation, each period t contains s sub-periods. For example, t and i could denote years and quarters with 4 s = (e.g., if quarterly observations are obtained by interpolating annual observations), or decades and years within decades, with 10 s = (e.g. if annual observations are obtained by interpolating decennial benchmark observations). We thus rewrite the series as x is the interpolated series, and ( ) g ψ is an indicator function which equals 1 if g is true and 0 otherwise, to ensure that MA padding is applied only to the interpolated observations. 4

Lagging (4) one-period, we obtain

( ) ( ) , 1 , 1, , 1 , 2 1 1 1 t i t s t s t i t i i s i x y y i s s ε θε ψ - - - - - -+ = + + - ≠ . (5)
One-period difference of the interpolated series equals

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , 1 , 1, , , 1 , 1 , 2 ,1 1, ,2 ,1 ,3 ,2 , 1 , 2 , , 1 , , 1 , 1 ,
ε θε ψ ε θε ψ ε θε ε θε ε θε ε θε ε θε ε θε ψ ε θε ψ - - - - - - - - - - - - -= - + - ≠ - - ≠   = - + - + - + + - + -       + - ≠ - - ≠     
Therefore, the short-variance of the interpolated series 2 1,x σ , as shown in the Appendix, equals:

( ) ( )( ) ( ) 2 1, , , 1
2 2 2 2 var 1 2 1 2 3 1 x t i t i x x s s s s s ε σ σ θ θ - = -     = + -+ -+       (6)
To compute the long-variance of the interpolated series 2 , s x σ , we start with s-period lag 4 In benchmark periods, the original and the interpolated series coincide by construction. i.e., in (4),

, , t i t i x y = when i s = . 1, 2, 1, 1 s t s t s t j j y y u - - - = = + ∑ . (7)
Then, the interpolated series equals

( ) ( ) ( ) ( ) ( ) ( ) , , 1, , , 1 1, , 1, , , 1 1 1
ε θε ψ ε θε ψ ε θε ψ - - - - - = - - = - = + + - ≠   - = + + + - ≠     = + + - ≠ ∑ ∑ Using (7), , t i x equals ( ) ( ) , 2, 1, , , , 1 1 1 s s t i t s t j t j t i t i j j i x y u u i s s ε θε ψ - - - = = = + + + - ≠ ∑ ∑ (8)
Lagging ( 8), we obtain

( ) ( ) 1, 2, 1, 1, 1, 1 1 s t i t s t j t i t i j i x y u i s s ε θε ψ - - - - -- = = + + - ≠ ∑ (9)
Using ( 8) and ( 9), the long difference of the interpolated series equals

( ) ( ) ( ) ( ) ( ) ( ) ( ) , 1, 1, , , , 1 1, 1, 1 1 1 1,1 2, 1,2 1,1 1,3 1,2 1, 1 1, 2 1, 1,
ε θε ε θε ψ ε θε ε θε ε θε ε θε ε θε ε θε - - - - -- = = - - - - - - -- -- - -- - -     - = + + - - + ≠         -    = - + - + - +       + - + -    + -     ∑ ∑  ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1, ,2 ,1 ,3 ,2 , 1 , 2 , , 1 , , 1 1, 1, 1 s t t t t t s t s t s t s t i t i t i t i i s ε θε ε θε ε θε ε θε ε θε ε θε ψ - - - - - --  + - + - +   + - + -  + - - + ≠  Therefore, the long variance of the interpolated series ( ) 2 , , 1, var s x t i t i x x σ - = - , as shown in the Appendix, equals ( ) ( ) ( ) ( ) 2 2 2 3 2 , 2 2 3 2 2 1 2 6 8 3 3 2 4 3( 1) 3 4 3 2 3 6 2 4 3( 1) 3 s x s s s s s s s s s s s s ε ε σ θ σ σ θ       +   + -+ + -+ +                    - - + -+ -+ +           = ( ) ( ) 2 2 2 3 2 3 2 2 2 1 2 6 6 4 3 17 24 3 3 s s s s s s s s ε ε σ σ θ θ       +   = + + -- - + -                 (10)
Thus, the variance-ratio for the interpolated series, using ( 6) and (10) and k s = , is given by

( ) ( ) ( )( ) ( ) 2 , 2 1, 2 2 2 3 2 3 2 2 2 2 2 2 2 1 2 6 6 4 3 17 24 3 3 1 2 1 2 3 1 s x x x V s s s s s s s s s s s s s s s ε ε ε σ σ σ σ θ θ σ θ θ =       +   + + -- - + -                =     + -+ -+       ( )( ) ( ) ( )( ) ( ) 2 3 2 3 2 2 2 2 1 2 6 6 4 3 17 24 3 1 2 1 6 3 1 θ θ θ + + + -- - + - = + -+ -+ (11)

Shock-persistence comparisons for original and interpolated series

A variance-ratio 2 2 , 1, Y s Y Y V s σ σ =
smaller than 1, suggests that the long-run component of the series as measured by 2 , s Y σ , is stable relative to year-to-year changes as measured by 2 1,Y σ .

Shock-persistence for the original series is given by equation ( 1), where ( )

2 1 0 s θ + > because 2 s ≥ and 1 1 θ -< < . The size of Y V therefore depends on the sign of θ : if 0 θ < , then 1 Y V > , and if 0 θ > , then 1 Y V < .
For a random-walk with white noise errors ( 0 θ = ) and no-drift, the variance ratio equals 1 as equation ( 1) shows. See Table 1 andFigure 1. Shock-persistence measure for the interpolated series, given by equation ( 11), depends on the MA parameter θ and on the interpolation segment length s which in this case also represents the timelength of the long-difference. To identify the impact of interpolation, we examine how shockpersistence varies with the two parameters θ and s, by numerically evaluating the variance-ratio of both the original and the interpolated series for various values of the parameters.

Table 1 exhibits the results. Several observations are noteworthy. First, variance-ratio of the original random-walk exhibits significant variation with respect to θ and s, varying from 0.03 (no shockpersistence), to 1.97 (significant shock-persistence). For all values of s, variance-ratio declines as θ increases. The decline is steeper for larger values of s. This directional consistency, however, is not reciprocal. For negative values of θ, variance-ratio increases with s, but for positive values of θ, variance-ratio decreases with s. For θ = 0, it is 1 for all values of s.

Second, the interpolated series has a smaller variance-ratio than the original series for each parameter value, confirming that interpolation reduces shock-persistence. This finding provides analytical support for [START_REF] Jaeger | Shock persistence and the measurement of prewar output series[END_REF] simulation estimates which, as noted, had little inferential value due to large standard errors of these estimates. Jaeger's simulations results reported in Table 1 (last column), and their comparable counterparts from our theoretical work reported in column 4, highlights this point. Third, while interpolation reduces the shock-persistence of a nonstationary series, the interpolated series may still exhibit significant shock-persistence. Thus, a low shock-persistence cannot be automatically attributed to interpolation. Indeed, [START_REF] Leung | Changes in the behavior of output in the United Kingdom, 1856-1990[END_REF] finds that the persistence differences between US and UK, which Jaeger uses to corroborate his findings, hold only for the particular UK output series that Jaeger used (NNP series). Alternative UK series (GDP at market price or factor cost) are similar in terms of shock-persistence to the interpolated prewar US output series. Thus, a low shockpersistence is not a sole artifact of interpolation.

Figure 1 displays the pattern of variance-ratio of the original series as data generating parameter θ and interpolation segment s change. As the figure shows, the variance-ratio may increase or decrease beyond the reference point of 1 as θ varies, if the series is a pure random-walk. The changes are more pronounced at higher values of s but more variable with respect to s at lower values of s. We also observe a prominent symmetry in the behavior of the variance-ratio for positive and negative values of θ, as s increases.

Figure 2 displays the pattern of variance-ratio of the interpolated series as we vary the DGP parameter θ and the interpolation segment s. Here we find that when 0 θ < , there are smaller changes in variance-ratio away from the reference point of 1 than when 0 θ > . Moreover, for positive values of θ, variance-ratio drops sharply with s at low values of s, then it remains steady as s increases. There is no symmetry in the behavior the variance ratio of the interpolated series. This is a further indication of the nonuniformity in the effect of linear interpolation on nonstationary series. These patterns reinforce our conclusion that interpolation affects data in complicated ways that go beyond simple rules.

Figure 3 shows the difference between the variance ratios of the original and interpolated series

y x V V
-as we vary the parameters θ and s. The difference is higher for negative values of θ than for positive values of θ . Further, the effect of increase in s is relatively sharp for lower values of s.

Fourth, as we show in the Appendix, the variance-ratio for the interpolated series exhibits periodic nonstationarity as the underlying moments are conditional on observation index i. 5 Using iterated expectation, we removed this conditionality, and thus eliminated the dependency on individual i's.

However, the presence of s in the shock-persistence measures we derive highlights this point.

In closing, we note a subtle statistical point. The variations and changes in the variance ratio parameter due to change in model parameters or interpolation length that we alluded to in the above discussions refer to the actual parameters derived from the DGP or its interpolated form. The sampling variation and uncertainty that comes with any inference about variance ratio parameter would depend on sample size and method of estimation and must not be confounded with the above theoretical results that are derived by changing the underlying model parameters.

Concluding Remarks

Stock and Watson (1986) suggest that linear interpolation of prewar U.S. macroeconomic series is the likely cause of the shock-persistence difference between these series and most European prewar or similar U.S. postwar series. [START_REF] Jaeger | Shock persistence and the measurement of prewar output series[END_REF] simulation results is a rare confirmation of this assertion, but the large variations in his simulation-based persistence measures detracts from its inference value.

We derive analytically the impact of interpolation on shock-persistence in Jaeger's model, and also go further to uncover a few additional interpolation-caused effects.

Our findings are as follows. First, linear interpolation reduces the shock-persistence of a randomwalk with MA(1) errors, confirming Stock and Watson's (1986) conjecture and [START_REF] Jaeger | Shock persistence and the measurement of prewar output series[END_REF] simulation results. Second, however, the interpolated random-walk series, may still exhibit significant shock persistence, with variance-ratio attaining values greater than 1, suggesting that interpolation is not synonymous with low shock persistence. Third, linear interpolation introduces periodic nonstationarity in a series. Fourth, the effect of linear interpolation on shock-persistence depends on the parameters of the underlying DGP as well as on the interpolation segment length.

Overall, our results suggest that using a simple rule to describe the effect of interpolation on shockpersistence would be an overreach even in a simple model such as [START_REF] Jaeger | Shock persistence and the measurement of prewar output series[END_REF]. That is because the determinants of this effect, which include the DGP parameters, the length of the interpolation segment, and the periodic nonstationarity that interpolation introduces, interact in complex ways. An important implication of this finding is that alternative causes of the difference in the persistence estimates for the U.S. prewar and postwar data merits further investigation.

We suspect that the results we report here are specific to the particular model we study and thus any statements about their generalizability should be made with caution. It is, therefore, important that future work examines the implications of linear interpolation for more general classes of models, and for both stationary and non-stationary series.

Table 1

Values of the variance-ratio for the original (non-interpolated) and interpolated series, for different values of θ and s. for the values of θ between -0.99 and 0.99, and for the values of s between 5 and 30.

2. The DGP is random walk with MA(1) errors with parameter θ and the length of the long-difference for the variance-ratio s. 3. The figures in the table were computed using equations ( 1) and ( 11), for the original and for the interpolated series, respectively. 4. The source of the figures presented in the last column is [START_REF] Jaeger | Shock persistence and the measurement of prewar output series[END_REF] V given in equation ( 1), for values of θ between -0.99 and 0.99, and for values of s between 3 and 30.

2. The DGP is random walk with MA(1) errors with parameter θ and the length of the long-difference for the variance-ratio s. 1

To obtain the variance ratio for the interpolated series (section 3 in the paper), we need to find ( )

2 1, , , 1 var x t i t i x x σ - = -
, the variance of the series' 1-period growth or the "short variance," and

( ) 2 , , 1, var s x t i t i x x σ - = -
, the variance of the series' s-period growth or the "long variance."

Short Variance

The short variance, i.e., the variance of the "short difference" can be obtained by taking the variance of the short difference in the interpolated series, as shown below.

( )

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) 2 1, , , 1 , 1, , , 1 , 1 , 2 ,1 1, ,2 ,1 ,3 ,2 , 1 , 2 , , 1 , , 1 , 1 , 2 var
σ ε θε ψ ε θε ψ ε θε ε θε ε θε ε θε ε θε ε θε ψ ε θε ψ - - - - - - - - - - - - = -   = - + - ≠ - - ≠     = - + - + - + + - + -     + - ≠     + - ≠  L ( ) ( ) ( ) ( ) ( ) ( ) { ( ) ( ) } ( ) ( ) ( ) ( ) ( ) ( ) { ( ) ( ) ,1 1, , 2 ,1 ,3 ,2 , 1 , 2 , , 1 , , 1
i ε θε ε θε ε θε ε θε ε θε ε θε ψ ε θε ε θε ε θε ε θε ε θε ε θε ψ - - - - - - - - - - -      + - + - + - + + - + -       - ≠       - - + - + - + + - + -      - ≠ L L } ( ) ( ) ( ) ( ) { } , , 1 , 1 , 2 2cov , 1
t i t i t i t i i s i ε θε ψ ε θε ψ - - -        + - ≠ - ≠    
Applying the definitions of variances and covariances, we obtain

( ) ( ) ( ) ( )( ) { ( )( ) } ( ) ( ) 2 2 1, , 1 1, , , 1 ,1 1, ,2 ,1 , 1 , 2 , , 1 , , 1 1 var
E i s σ ε θε ε θε ε θε ε θε ε θε ε θε ε θε ψ - - - - - - -   = - + + - + - - +     + - -     + - ≠     ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { ( ) ( ) } ( ) ( ) ( ) ( ) ( ) ( ) , 1 , 2 ,1 1, ,2 ,1 ,3 ,2 , 1 , 2 , , 1 , , 1 ,1 1, ,2 ,1 ,3 ,2 , 1 , 2 , , 1 var 1
ε θε ψ ε θε ε θε ε θε ε θε ε θε ε θε ψ ε θε ε θε ε θε ε θε ε θε - - - - - - - - - - -   + - ≠       + - + - + - + + - + -       - ≠     - - + - + - + + - + -    { ( ) ( ) } ( ) ( ) ( ) ( ) { } , 1 , 2 , , 1 , 1 , 2
, 1 2 cov , 1

t i t i t i t i t i t i i i s i ε θε ψ ε θε ψ ε θε ψ - - - - -        - ≠       + - ≠ - ≠    
which can be simplified

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 1, 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 1 2 1, 2 2 2 2 1, 2 2 2 1 2 2 2 x s s s i s i i i s i i s s s s s i s s s s s s ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε σ σ θ σ θσ σ θ σ ψ σ θ σ ψ θσ ψ σ θσ θσ θ σ ψ σ θσ θ σ ψ σ θσ θ σ     = + + -- + + ≠         + + ≠ + ≠ ≠         + - - + ≠ ≠             + - + =           + - + -     ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 i s i s s s s ε ε ε ε ψ σ θσ θσ θ σ ψ   =         + - + + - ≥ >        
Collecting terms, we obtain:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 1, 2 2 2 2 2 2 2 2 2 2 2 2 1 2 1 1 1 1 2 1, 2 4 2 1, 2 2 2 1 2 2 2 2 2 4 2 2 x i s s i s s i i i s i i s i s s i s i s s ε ε ε ε ε ε ε ε σ σ θ θ σ θ ψ σ θ ψ σ θ ψ σ σ θ θ ψ θ θ ψ σ σ θ θ ψ θ θ ψ       = + - - + + ≠             + + ≠ + ≠ ≠             + -+ ≠ ≠ + -+ =                     + -+ - = + -+ - ≥ >             ψ ψ ψ ψ ψ ψ = = = = = = - ≠ = ≠ = = = = = - ≠ ≠ = ≥ > = ∑ ∑ ∑ ∑ θ θ θ θ θ θ     = + - -        - - -       + + + + +                   -     + -+ + -+               -     + -+ - + -+ -        
which is the unconditional short variance of the interpolated series. Simplifying the equation, after collecting terms, we obtain

( )( ) ( ) 2 2 2 2 1, 2 1 2 1 2 3 1 x s s s s s ε σ σ θ θ     = + -+ -+      
This is the expression for the short variance that is given in equation ( 6) in the paper.

Long Variance

The long variance, i.e., the variance of the "long difference" can be obtained by taking the variance of the long difference in the interpolated series, as shown below.

(

( ) ( ) ( ) ( ) ( ) ( ) 2 , , 1, 1, , , , 1 1, 1, 1 1 1 2 1,1 2, 1,2 1,1 1, 1, 1 2 ,1 )
i s σ ε θε ε θε ψ ε θε ε θε ε θε ε θε ε - - - - -- = = - - - - - -- - = -   -     = + + - - + ≠             -     = - + - + + -         + - +     ∑ ∑  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ,1 , , 1 , , 1 1, 1, 1 1,1 2, 1,2 1,1 1, 1, 1 ,1 1, , 2 ,1 ,
ε θε ε θε ε θε - - - -- - - - - - -- - -   - + + -     + - - + ≠    -     + - + - + + -              - + - + + -            ( ) ( ) ( ) ( ) { ( ) ( ) } ( ) ( ) ( ) ( ) { ( ) ( ) } 1,1 2, 1,2 1,1 1, 1, 1 , , 1 1, 1, 1 ,1 1, , 2 ,1 , , 1 ,
ε θε ε θε ε θε ε θε ε θε ψ ε θε ε θε ε θε ε θε ε θε ψ - - - - - -- - - -- - - - - --   + - - + - + + -         - - + ≠     + - + - + + -     - - + ≠     which yields i i i s s ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε σ σ θ σ θσ σ θ σ θσ σ θ σ σ θ σ ψ θσ θσ σ θσ θ σ ψ σ θσ θ σ ψ                                                  - = + + -- + + + -- + + + + ≠ - + - - + - -+ - = + -+ - ≠ ≠ ( ) ( ) ( ) (           +                                                   - = + - - - + + ≠ - - + -- = + -+ - ≠ ≠ + -+ = + -+ ≠ ≠
where we use the notation ( ) 2 , s x i σ to emphasize that the long variance of the interpolated series also depends on i. Following the same steps as above to remove the conditionality of the long variance on i, and using the summation for the indicator functions as we did above in this Appendix, we obtain

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) σ θ σ θ σ = = =   -                 ≠                        -- = + -+ - ≠ ≠                            -       + = + - - - + + - - + + ∑ ∑ ∑ ( ) ( ) ( ) ( ) 2 2 1 1 1 1 2 1 2 4 2 1, s s i i i i i s s s θ θ ψ θ θ ψ = =           + = + -+ ≠ ≠                     ∑ ∑ where 1 1 1 2 s i s i s i = +   =     = ∑ and ( )( ) 2 2 1 1 2 1 1 6 s i s s i i s = + +   = =     ∑ After the substitution, we have [ ] [ ] ( ) ( ) ( ) [ ] [ ] [ ] ( ) ( )   - + + + +     + - -           + -+ +   + + -           -+  -     + -- + -+ -                 +   -     = - + ( ) ( ) σ θ θ σ       +   + + -- - + -                 =
This is the expression for the long variance that is given in equation ( 10) in the paper.

Mathematica commands for plotting Figures 1, 2, and3 Figure 1 Plot

[{( 𝑠𝑠(1 + (-0.99) 2 ) + �2(1 -𝑠𝑠)(-0.99)� 𝑠𝑠(1 + (-0.99) 2 ) ), � 𝑠𝑠(1 + (-0.75) 2 ) + �2(1 -𝑠𝑠)(-0.75)� 𝑠𝑠(1 + (-0.75) 2 ) �, � 𝑠𝑠(1 + (-0.5) 2 ) + �2(1 -𝑠𝑠)(-0.5)� 𝑠𝑠(1 + (-0.5) 2 ) �, � 𝑠𝑠(1 + (-0.25) 2 ) + �2(1 -𝑠𝑠)(-0.25)� 𝑠𝑠(1 + (-0.25) 2 ) �, � 𝑠𝑠(1 + (0) 2 ) + �2(1 -𝑠𝑠)(0)� 𝑠𝑠(1 + (0) 2 ) �, � 𝑠𝑠(1 + (0.25) 2 ) + �2(1 -𝑠𝑠)(0.25)� 𝑠𝑠(1 + (0.25) 2 ) �, � 𝑠𝑠(1 + (0.5) 2 ) + �2(1 -𝑠𝑠)(0.5)� 𝑠𝑠(1 + (0.5) 2 ) �, � 𝑠𝑠(1 + (0.75) 2 ) + �2(1 -𝑠𝑠)(0.75)� 𝑠𝑠(1 + (0.75) 2 ) �, ( 𝑠𝑠(1 + (0.99) 2 ) + �2(1 -𝑠𝑠)(0.99)� 𝑠𝑠(1 + (0.99) 2 ) )},
{𝑠𝑠, 2,30}, PlotLabels → {"Teta = -0.99", "Teta = -0.75", "Teta = -0.50", "Teta = -0.25", "Teta = 0.00", "Teta = 0.25", "Teta = 0.50", "Teta = 0.75", "Teta = 0.99"}, PlotTheme-

> "Monochrome"] Figure 2 Plot[{ (1 + (-0.99) 2 )(2𝑠𝑠 3 + 6𝑠𝑠 2 + 𝑠𝑠 -6) -(-0.99)(4𝑠𝑠 3 -3𝑠𝑠 2 + 17𝑠𝑠 -24) 3𝑠𝑠 2 (1 + (-0.99) 2 )(2𝑠𝑠 -1) + 6𝑠𝑠(-0.99)(𝑠𝑠 2 -3𝑠𝑠 + 1) , (1 + (-0.75) 2 )(2𝑠𝑠 3 + 6𝑠𝑠 2 + 𝑠𝑠 -6) -(-0.75)(4𝑠𝑠 3 -3𝑠𝑠 2 + 17𝑠𝑠 -24) 3𝑠𝑠 2 (1 + (-0.75) 2 )(2𝑠𝑠 -1) + 6𝑠𝑠(-0.75)(𝑠𝑠 2 -3𝑠𝑠 + 1) , (1 + (-0.5) 2 )(2𝑠𝑠 3 + 6𝑠𝑠 2 + 𝑠𝑠 -6) -(-0.5)(4𝑠𝑠 3 -3𝑠𝑠 2 + 17𝑠𝑠 -24) 3𝑠𝑠 2 (1 + (-0.5) 2 )(2𝑠𝑠 -1) + 6𝑠𝑠(-0.5)(𝑠𝑠 2 -3𝑠𝑠 + 1) , (1 + (-0.25) 2 )(2𝑠𝑠 3 + 6𝑠𝑠 2 + 𝑠𝑠 -6) -(-0.25)(4𝑠𝑠 3 -3𝑠𝑠 2 + 17𝑠𝑠 -24) 3𝑠𝑠 2 (1 + (-0.25) 2 )(2𝑠𝑠 -1) + 6𝑠𝑠(-0.25)(𝑠𝑠 2 -3𝑠𝑠 + 1) , (1 + (0) 2 )(2𝑠𝑠 3 + 6𝑠𝑠 2 + 𝑠𝑠 -6) -(0)(4𝑠𝑠 3 -3𝑠𝑠 2 + 17𝑠𝑠 -24)
3𝑠𝑠 2 (1 + (0) 2 )(2𝑠𝑠 -1) + 6𝑠𝑠(0)(𝑠𝑠 2 -3𝑠𝑠 + 1) ,

(1 + (0.25) 2 )(2𝑠𝑠 3 + 6𝑠𝑠 2 + 𝑠𝑠 -6) -(0.25)(4𝑠𝑠 3 -3𝑠𝑠 2 + 17𝑠𝑠 -24) 3𝑠𝑠 2 (1 + (0.25) 2 )(2𝑠𝑠 -1) + 6𝑠𝑠(0.25)(𝑠𝑠 2 -3𝑠𝑠 + 1) ,

(1 + (0.5) 2 )(2𝑠𝑠 3 + 6𝑠𝑠 2 + 𝑠𝑠 -6) -(0.5)(4𝑠𝑠 3 -3𝑠𝑠 2 + 17𝑠𝑠 -24) 3𝑠𝑠 2 (1 + (0.5) 2 )(2𝑠𝑠 -1) + 6𝑠𝑠(0.5)(𝑠𝑠 2 -3𝑠𝑠 + 1) ,

(1 + (0.75) 2 )(2𝑠𝑠 3 + 6𝑠𝑠 2 + 𝑠𝑠 -6) -(0.75)(4𝑠𝑠 3 -3𝑠𝑠 2 + 17𝑠𝑠 -24) 3𝑠𝑠 2 (1 + (0.75) 2 )(2𝑠𝑠 -1) + 6𝑠𝑠(0.75)(𝑠𝑠 2 -3𝑠𝑠 + 1) ,

(1 + (0.99) 2 )(2𝑠𝑠 3 + 6𝑠𝑠 2 + 𝑠𝑠 -6) -(0.99)(4𝑠𝑠 3 -3𝑠𝑠 2 + 17𝑠𝑠 -24) 3𝑠𝑠 2 (1 + (0.99) 2 )(2𝑠𝑠 -1) + 6𝑠𝑠(0.99)(𝑠𝑠 2 -3𝑠𝑠 + 1) }, {𝑠𝑠, 2,30}, PlotLabels → {"Teta = -0.99", "Teta = -0.75", "Teta = -0.50", "Teta = -0.25", "Teta = 0.00", "Teta = 0.25", "Teta = 0.50", "Teta = 0.75", "Teta = 0.99"}, PlotTheme-> "Monochrome"] -(1 + (0.99) 2 )(2𝑠𝑠 3 + 6𝑠𝑠 2 + 𝑠𝑠 -6) -(0.99)(4𝑠𝑠 3 -3𝑠𝑠 2 + 17𝑠𝑠 -24) 3𝑠𝑠 2 (1 + (0.99) 2 )(2𝑠𝑠 -1) + 6𝑠𝑠(0.99)(𝑠𝑠 2 -3𝑠𝑠 + 1) )}, {𝑠𝑠, 2,30}, PlotLabels → {"Teta = -0.99", "Teta = -0.75", "Teta = -0.50", "Teta = -0.25", "Teta = 0.00", "Teta = 0.25", "Teta = 0.50", "Teta = 0.75", "Teta = 0.99"}, PlotTheme-> "Monochrome"]

Fig. 1 .

 1 Fig. 1. Effects of changing the values of θ and s on the variance-ratio of the original series

Fig. 2 .

 2 Fig. 2. Effects of changing the values of θ and s on the variance-ratio of the interpolated series Notes: The figure shows the values of the variance-ratio of the interpolated series x V given in equation (11). See the

Fig. 3 .

 3 Fig. 3. The difference between the values of the variance ratio of the original and interpolated series, for different values of θ and s

  Figure3

3. Shock-persistence of interpolated random-walk with MA(1) errors

	To model interpolated series, we divide the original time series T Y into segments of equal length s,
	drop all but one observation within each segment, and reconstruct the "missing" observations by
	linearly interpolating the remaining observations. To facilitate the conversion, rewrite T Y as , t i y ,
	where	t =	0,1, 2,...	and	i	=	1, 2,...,	s	, where	2 s ≥ . Then, 1	0,1

The use of interpolation is not limited to the prewar data. For example,[START_REF] Levy | Estimates of the aggregate quarterly capital stock series for the post-war U.S. economy[END_REF] employ the method of linear interpolation to construct quarterly values of the US postwar capital stock and capital stock depreciation series using the series' annual observations.[START_REF] Levy | Not all price endings are created equal: Price points and asymmetric price rigidity[END_REF] apply linear interpolation to weekly retail scanner price data during the 1989-1997 period, to determine the values of missing observations.

Examples include DeLong and Summers (1988),[START_REF] Campbell | Are output fluctuations transitory?[END_REF],Stock and Watson (1986),[START_REF] Cecchetti | Variance-ratio tests: Small-sample properties with an application to international output data[END_REF],[START_REF] Murray | The uncertain trend in U.S. GDP[END_REF], and[START_REF] Charles | A note on the uncertain trend in US real GNP[END_REF], among others.

Interpolation, however, has consequences for other issues as well, including dating of business cycles[START_REF] Charles | A revision of the US business-cycles chronology, 1790-1928[END_REF], classification of the 19 th century inflation[START_REF] Kaufmann | Is deflation costly after all? The perils of erroneous historical classifications[END_REF], transmission of monetary policy in the EU[START_REF] Ehrmann | Comparing monetary policy transmission across European countries[END_REF], data periodicity[START_REF] Franses | Data revisions and periodic properties of macroeconomic data[END_REF], and chaotic dynamics (Orlando and Zimatore 2018).

[START_REF] Dezhbakhsh | Periodic properties of interpolated time series[END_REF] derive the variance, the covariance, and the autocorrelation functions of linearly interpolated trend-stationary series, and find that they all vary with i, which they term "periodic variation."

[START_REF] Dezhbakhsh | Periodic properties of interpolated time series[END_REF] derive the variance, the covariance, and the autocorrelation functions of linearly interpolated trend-stationary series, and find that they all vary with i, which they term "periodic variation."

2

Monetary Economics 10, 139-162. Orlando, G., Zimatore, G., 2018