Nadia Boussaha

Fayçal Hamdi
email: fhamdi@usthb.dz

Saïd Souam

Multivariate Periodic Stochastic Volatility Models: Applications to Algerian dinar exchange rates and oil prices modeling

Keywords: Multivariate periodic stochastic volatility, periodic stationarity, periodic Kalman filter, particle filtering, exchange rates, Saharan Blend oil

The contribution of this paper is twofold. In a first step, we propose the so called Periodic Multivariate Autoregressive Stochastic Volatility (P V AR-SV ) model, that allows the Granger causality in volatility in order to capture periodicity in stochastic conditional variance. After a thorough discussion, we provide some probabilistic properties of this class of models. We thus propose two methods for the estimation problem, one based on the periodic Kalman filter and the other on the particle filter and smoother with Expectation-Maximization (EM ) algorithm. In a second step, we propose an empirical application by modeling oil price and three exchange rates time series. It turns out that our modeling gives very accurate results and has a well volatility forecasting performance.

Introduction

Instantaneous volatility and volatility clustering modeling play an important role in the analysis of financial time series. Since the introduction of the Autoregressive Conditional Heteroskedasticity (ARCH) model in the seminal paper of [START_REF] Engle | Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation[END_REF], a significant part of the literature was devoted to these issues. However and despite its success, this modeling exhibits some drawbacks. A satisfactory alternative to ARCH-type models was introduced by [START_REF] Taylor | Financial returns modeled by the product of two stochastic processes -a study of daily sugar prices 1961-79[END_REF]. The namely stochastic volatility (SV ) model allows the variance of the returns to be an unobserved random process. This is not the case in ARCH and their generalized GARCH models, where volatility is a function of previous observations and/or past volatility. Moreover, the SV model allows the log of the volatility to evolve. This ensures that the variance of the process always remains positive without the need for further constraints, as is the case in the ARCH/GARCH models.

Apart from the volatility clustering phenomenon, there are other important stylized facts associated with financial returns series. One can cite excess kurtosis, asymmetry and heavy-tailed errors, persistence (long memory property), etc. In practice, it turns out that a large part of the literature was based on univariate models. However, some stylized facts cannot be captured by a univariate description. This is, for example, the case of the covariation effect, i.e. the study of the relationships between the volatilities and covolatilities of several markets. On one hand, much of the financial decision making (such as portfolio optimization, asset allocation, risk management, and asset pricing) clearly needs to take correlations into account. On the other hand, it is now well documented that financial volatilities of different assets and markets move together over time. Large changes in one asset are matched by large movements in another.

This fact plays a critical role in the construction of the appropriate modeling of financial time series. The multivariate models for modeling timevarying volatility are particularly interesting. Indeed, they can lead to greater statistical efficiency. As a result, working with multivariate modeling framework leads to more relevant empirical models than with separate univariate models. Some multivariate stochastic volatility models have even recently become a major concern in the investigation of the correlation structure of multivariate economic series in general, and of multivariate financial time series in particular. Various extensions of the basic Multivariate SV (M SV ) models have been proposed in the literature. One can cite for example [START_REF] Harvey | Multivariate stochastic variance models[END_REF], [START_REF] Daníelsson | Multivariate stochastic volatility models: estimation and a comparison with V GARCH models[END_REF], [START_REF] Jungbacker | Monte Carlo likelihood estimation for three multivariate stochastic volatility models[END_REF], [START_REF] Smith | Foreign exchange intervention by the Bank of Japan: Bayesian analysis using a bivariate stochastic volatility model[END_REF], [START_REF] Chan | Multivariate stochastic volatility models with correlated errors[END_REF], Asai and McAleer (2009a, b), etc. [START_REF] Asai | Multivariate stochastic volatility: a review[END_REF] propose a detailed review of the literature. Note that these models have attracted a lot of attention in modern finance theory and enjoyed voluminous empirical applications. [START_REF] Yu | Multivariate stochastic volatility models: Bayesian estimation and model comparison[END_REF] thoroughly discuss a wide range of dominant M SV models, available in the literature, on specification, estimation, and evaluation of M SV models. It is worth noting that all of the proposed models deal with constant volatility parameters. There seems to be no formulation that can adequately explain multivariate time series whose structure changes over time. This is, in particular, the case for time series with a volatility displaying a periodic correlation pattern. For instance, it is usually observed in financial time series for which correlations between daily returns and volatility display some day-of-the-week effects (see e.g. [START_REF] Franses | Modelling day-of-the-week seasonality in the S&P 500 index[END_REF], [START_REF] Bentarzi | Mixture Periodic Autoregressive Conditional Heteroskedastic Models[END_REF] and [START_REF] Hamdi | Mixture Periodic GARCH Models: Theory and Applications[END_REF]). Hence, it is interesting to assume that the log-volatilities in each day (more generally, season) might be described by a different model. More specifically, we can assume that the parameters change periodically over time.

The contribution of our paper is twofold. We, firstly, propose a model, called Periodic Multivariate Autoregressive Stochastic Volatility (P V AR-SV ) model, that allows the Granger causality in volatility in order to capture periodicity in stochastic conditional variance. We, secondly, provide an empirical application which shows that our modeling gives very accurate results and has a well volatility forecasting performance.

The remainder of the paper is organized as follows. Section 2 thoroughly describes the P V AR-SV models. In Section 3, we provide some probabilistic properties of this class of models. In Section 4, we propose two straightforward methods for the estimation problem, for which the implementation is based on periodic Kalman filter and on the particle filter and smoother with expectation-maximization (EM ) algorithm. Finally, in Section 5 we apply our model to a set of three exchange rates and oil price time series, namely U.S. dollar/Algerian dinar (U SD/DZD), Euro/Algerian dinar (EU R/DZD), Euro/U.S. dollar (EU R/U SD) and the daily Saharan Blend oil prices.

Multivariate Periodic Autoregressive Stochastic Volatility

The univariate periodic autoregressive stochastic volatility model (P AR-SV ) was introduced by Aknouche et al. (2007) and thoroughly studied by [START_REF] Boussaha | On Periodic Autoregressive Stochastic Volatility Models[END_REF]. It provides a successful alternative to the class of periodic generalized autoregressive conditionally heteroscedastic (P GARCH) models [START_REF] Bollerslev | Periodic autoregressive conditional heteroscedasticity[END_REF]. Indeed, it takes into account the time-varying and persistent volatility as well as the periodicity feature in the autocorrelation structure exhibited by many nonlinear time series.

A stochastic process {ε t ; t ∈ Z} has a P AR-SV representation of period S, if it is a solution of the following stochastic difference equation

{ ε t = η t exp { 1 2 x t } , x t = α t + β t x t-1 + Q t e t , t ∈ Z, (1)
where the parameters α t , β t and Q t are periodic in t with period S (i.e., α t+τ S = α t , β t+τ S = β t and Q t+τ S = Q t , for all t, τ ∈ Z). The two independent sequences (η t ) and (e t ) are independent and identically distributed (i.i.d.) random variables with zero mean and unit variance.

In the current paper, we propose a multivariate periodic autoregressive stochastic volatility (P V AR-SV ) model. Let ε t = (ε t,1 , ..., ε t,m ) ′ be the m × 1 vector of stock returns at time t and X t = (X t,1 , ..., X t,m ) ′ be the corresponding vector of log-variances. The P V AR-SV model can be defined as follows

{ ε t = V 1/2 t η t , X t = α t + Φ t X t-1 + e t , t ∈ Z, (2)
where To emphasize the periodicity, let t = s + τ S, for τ ∈ Z and 1 ≤ s ≤ S. Then, model (2) can be written in the following form

V t = diag (exp X t,1 , ..., exp X t,m ) = diag (exp X t ), Φ t = ( ϕ (t) i,j ) i,j=1,...,
{ ε s+τ S = diag ( exp { 1 2 X s+τ S }) η s+τ S , X s+τ S = α s + Φ s X s+τ S-1 + e s+τ S , τ ∈ Z, 1 ≤ s ≤ S.
(3)

Periodic stationarity and computation of moments

Let us now investigate some of the basic properties of the P V AR-SV model (2). For the development of statistical estimation and testing theory, we need to provide conditions for periodic strict stationarity and periodic ergodicity (in the sense given, for example, by [START_REF] Aknouche | Quasi-maximum likelihood estimation of periodic GARCH and periodic ARM A-GARCH processes[END_REF]) of a multivariate P AR-SV process. It is clear from the multiplicative form of the second equation in model ( 2) that the process {X t ; t ∈ Z} is strictly periodically stationary and periodically ergodic if and only if the spectral radius of the matrix ∏ S v=1 Φ v is less than one. In order to go further, let us introduce some notation. Let A ⊗r = A ⊗ A ⊗ . . . ⊗ A be the r-th Kronecker power of any matrix A, where r ∈ {0, 1, ...} (by convention A ⊗0 = I and A ⊗1 = A). Denote by ρ (A) the spectral radius of any square matrix A.

It may be noted that if ρ

( ∏ S v=1 Φ v ) < 1, then {X t ; t ∈ Z} is a periodic stationary process with the unconditional periodic mean, for τ ∈ Z and s ∈ {1, 2, ..., S}, µ (s) X := E (X s+τ S ) = ( I m - S-1 ∏ v=0 Φ s-v ) -1 S-1 ∑ v=0 ( v-1 ∏ i=0 Φ s-i ) α s-v ,
and the unconditional periodic second order moment

vec ( Σ (s) X ) := E ( X ⊗2 s+τ S ) =   I m 2 - ( S-1 ∏ v=0 Φ s-v ) ⊗2   -1 S-1 ∑ v=0 ( v-1 ∏ i=0 Φ s-i ) ⊗2 vec ( Σ (s-v) e ) ,
where vec (A) denotes an n 1 n 2 -vector, obtained from an (n 1 × n 2 )-matrix A by stacking its columns in the natural order. If ρ ( ∏ S v=1 Φ v ) < 1, the process X t is given as a first order periodic V AR causal model which can be represented as

X s+τ S = µ (s) X + ∑ l≥0 ( l-1 ∏ i=0 Φ s-i ) e s+τ S-l .
Since ε t is the product of two strictly periodically stationary processes, it must also be strictly periodically stationary. Thus the conditions needed to ensure the periodic stationarity of ε t (in the strict and weak senses), are just the ones needed to ensure periodic stationarity of the process X t . This is summarized in the following theorem.

Theorem 1. Model (2) admits a unique nonanticipative (future-independent) strictly periodically stationary solution given, for τ ∈ Z and s ∈ {1, 2, ..., S}, by

ε s+τ S = diag ( exp { 1 2 ( µ (s) X + ∑ l≥0 ( l-1 ∏ i=0 Φ s-i ) e s+τ S-l )}) η s+τ S , ( 4
)
where the series in (4) converges almost surely, if and only if ρ

( S ∏ v=1 Φ v ) < 1. ( 5
)
In the sequel, we explicitly characterize the variance, skewness and kurtosis of the P V AR-SV model (2) . Indeed, it turns out that these moments are very useful to describe data. This is why they are widely used by most researchers.

Suppose that (5) holds and the two vectors η t and e t are multivariate normal. In a first step, we provide very general results. In a second step, we do consider normality in order to get more explicit formula for these moments. We thus have:

E ( ε ⊗r t ) = E ( ( V 1/2 t η t ) ⊗r ) = E [ diag { ( exp 1 2 X t ) ⊗r } × η ⊗r t ] = diag { E [ ( exp 1 2 X t ) ⊗r ]} E [ η ⊗r t ] = diag { E [ Y (r) t ]} E [ η ⊗r t ] ,
where

Y (r) t = ( exp 1 2 X t ) ⊗r = ( Y (r) t,1 , Y (r) t,2 , ..., Y (r) t,m r ) , with Y (r) t,i = exp ( 1 2 r-1 ∑ n=0 X t,k i,n
)

, for i = 1, ..., m r , with i -k i,0 = (k i,1 -1) m + • • • + (k i,r-1 -1) m r-1 ,
and

k i,0 = { m if i ≡ 0 ( mod m) , i ≡ 0 ( mod m) otherwise. From this last equation, it turns out that E ( Y (r) t,i ) = E ( exp { T ′ i,r X t }) = M Xt (T i,r ) , i = 1, ..., m r where M Xt (T ) = E (exp {T ′ X t }) denotes the moment generating function of X t , T i,r = 1 2 ∑ r-1 n=0 w k i,n
, and w j is the m × 1 canonical vector w j = ( 0 1×(j-1) , 1, 0 1×m-j

) ′ . Thus,

E ( ε ⊗r t ) = diag {(M Xt (T 1,r ) , ..., M Xt (T m r ,r ))} E [ η ⊗r t ] = diag {( T ′ 1,r µ (t) X + 1 2 T ′ 1,r Σ (t) X T 1,r , ..., T ′ m r ,r µ (t) X + 1 2 T ′ m r ,r Σ (t) X T m r ,r )} × E [ η ⊗r t ] .
Given the matrix expressions of the third and fourth (r = 3 and 4) moments of the P V AR-SV process {ε t ; t ∈ Z}, we can derive the exact matrix expression formulae of the skewness and kurtosis measures. These expressions are only functions of the parameters of the model. Let us note that non-linearities are typically analyzed through these pairwise measures in finance. They are defined as follows:

sk (t) ε := E ( [ ( Σ (t) ε,0 ) -1/2 ε t ] ⊗3 ) ′ E ( [ ( Σ (t) ε,0 ) -1/2 ε t ] ⊗3 )
, and

κ (t) ε := vec (I m 2 ) ′ E ( [ ( Σ (t) ε,0 ) -1/2 ε t ] ⊗4 )
, where

( Σ (t) ε,0
) 1/2 is any symmetric square root of the variance-covariance matrix of ε t (see e.g. [START_REF] Kollo | Multivariate skewness and kurtosis measures with an application in ICA[END_REF].

In the P V AR-SV case, we have

E ( [ ( Σ (t) ε,0 ) -1/2 ε t ] ⊗3 ) = ( ( Σ (t) ε,0 ) -1/2 ) ⊗3 × diag {( T ′ 1,3 µ (t) X + 1 2 T ′ 1,3 Σ (t) X T 1,3 , ..., T ′ m 3 ,3 µ (t) X + 1 2 T ′ m 3 ,3 Σ (t) X T m 3 ,3 )} E [ η ⊗3 t ] = 0 m 3 ×1 , since E [ η ⊗3 t ] = 0 m 3 ×1 , and E ( [ ( Σ (t) ε,0 ) -1/2 ε t ] ⊗4 ) = [ ( Σ (t) ε,0 ) -1/2 ] ⊗4 × diag {( T ′ 1,4 µ (t) X + 1 2 T ′ 1,4 Σ (t) X T 1,4 , ..., T ′ m 4 ,4 µ (t) X + 1 2 T ′ m 4 ,4 Σ (t) X T m 4 ,4 )} E [ η ⊗4 t ] .
Note that under the normality hypothesis, the closed form of E

[ η ⊗r t ]
can be obtained from the result provided by Kollo and von Rosen (2006, Corollary 2.2.7.4). Indeed, odd moments of η t are equal to zero and even moments are given by the following equalities:

E ( η ⊗2 t ) = vec ( Σ (t) η ) , E ( η ⊗4 t ) = (I m 4 + I m ⊗ K m,m ⊗ I m + I m ⊗ K m 2 ,m ) [ vec ( Σ (t) η )] ⊗2 ,
and

E ( η ⊗r t ) = r ∑ i=2 ( I m ⊗ K m i-2 ,m ⊗ I m r-i ) [ vec ( Σ (t) η ) ⊗ E ( η ⊗r-2 t )]
, for r = 2, 4, 6, ...

where K p,q is the pq × pq commutation matrix. The following proposition summarizes the exact general formula of the different moments and gives the variance, the skewness and the kurtosis of the P V AR-SV model.

Proposition 2. For a periodic stationary solution {ε t ; t ∈ Z} of the P V AR-SV model defined by (2), for any positive integer r and under the normality hypothesis of the two vectors η t and e t ,we have

µ (s) ε r := E ( ε ⊗r s+τ S ) = diag {( T ′ 1,r µ (s) X + 1 2 T ′ 1,r Σ (s) X T 1,r , ..., T ′ m r ,r µ (s) X + 1 2 T ′ m r ,r Σ (s) X T m r ,r )} × E [ η ⊗r s+τ S
] .

Furthermore, the variance, skewness and the kurtosis of the distribution of ε s+τ S are given by:

vec ( Σ (s) ε,0 ) := E ( ε ⊗2 s+τ S ) = diag {( T ′ 1,2 µ (s) X + 1 2 T ′ 1,2 Σ (s) X T 1,2 , ..., T ′ m 2 ,2 µ (s) X + 1 2 T ′ m 2 ,2 Σ (s) X T m 2 ,2 )} × vec ( Σ (s) η ) , sk (s) ε = 0, and
κ (s) ε := vec (I m 2 ) ′ [ ( Σ (s) ε,0 ) -1/2 ] ⊗4 × diag {( T ′ 1,4 µ (s) X + 1 2 T ′ 1,4 Σ (s) X T 1,4 , ..., T ′ m 4 ,4 µ (s) X + 1 2 T ′ m 4 ,4 Σ (s) X T m 4 ,4
)}

× (I m 4 + I m ⊗ K m,m ⊗ I m + I m ⊗ K m 2 ,m ) [ vec ( Σ (s) η )] ⊗2 .

The autocorrelation functions of the P V AR-SV model

In this section, we investigate the dependence structure in a periodic stationary solution {ε t ; t ∈ Z} of (2). We, more particularly, examine its autocovariance function, most frequently used by statisticians, time series analysts and practitioners.

Let us first recall that the periodic autocovariance function of a centered m-variate periodically stationary process {ε t ; t ∈ Z} is defined by

Σ (s) ε,h := cov (ε s+τ S , ε s+τ S-h ) = E ( ε t ε ′ t-h ) , h = 0, 1, 2, ...
It is more convenient, in practice, to work with the vector autocovariance function defined by vec

( Σ (s) ε,h ) := E (ε t ⊗ ε t-h ) .
The following proposition completely characterizes this vector autocovariance function.

Proposition 3. Let {ε t ; t ∈ Z} be a periodic stationary solution of (2). Then {ε t ; t ∈ Z} is a periodic weak white noise process with variance given, for s = 1, ..., S, by vec

( Σ (s) ε,0 ) = diag {( T ′ 1,2 µ (s) X + 1 2 T ′ 1,2 Σ (s) X T 1,2 , ..., T ′ m 2 ,2 µ (s) X + 1 2 T ′ m 2 ,2 Σ (s) X T m 2 ,2 )} × vec ( Σ (s) η ) .
Proof. For h > 0, we have

E [ε t ⊗ ε t-h ] = E [( diag ( exp { X t 2 }) η t ) ⊗ ( diag ( exp { X t-h 2 }) η t-h )] = E [ diag {( exp { X t 2 }) ⊗ ( exp { X t-h 2 })} (η t ⊗ η t-h ) ] = diag ( E [ exp { X t 2 } ⊗ exp { X t-h 2 }]) × E (η t ⊗ η t-h ) = diag ( E [ Y (1) t ⊗ Y (1) t-h ]) × E (η t ) ⊗ E (η t-h ) = diag ( E [ exp { X t,1 +X t-h,1 2 , ..., X t,1 +X t-h,m 2 , ..., Xt,m+X t-h,1 2 , ..., Xt,m+X t-h,m 2 }]) × E (η t ) ⊗ E (η t-h ) = 0 m 2 ×1 , since (η t ) is a centered sequence and X t,i and X t-h,j are two Gaussian ran- dom variables for all i, j = 1, ..., m, then E [ exp { X t,i +X t-h,j 2 }] < ∞, ∀i, j = 1, ..., m.
This defines the multivariate periodic AR-SV framework, in which there is no linear dependence structure in {ε t ; t ∈ Z}. Therefore, even though {ε t ; t ∈ Z} is a periodic white noise process, it is still possible that non-linear forms of dependence between the successive terms of {ε t ; t ∈ Z} exist. The nature of the dependence structure in the periodic V AR-SV process defined in (2) can be obtained by studying the covariance structure of the observations raised to the power r ∈ N * . We have from ( 2)

E [ ε ⊗2r t ⊗ ε ⊗2r t-h ] = E [ ( diag ( exp { Xt 2 }) η t ) ⊗2r ⊗ ( diag ( exp { X t-h 2 }) η t-h ) ⊗2r ] = diag ( E [ ( exp { Xt 2 }) ⊗2r ⊗ ( exp { X t-h 2 }) ⊗2r ]) × E ( η ⊗2r t ) ⊗ E ( η ⊗2r t-h ) = diag ( E [ Y (2r) t ⊗ Y (2r) t-h ]) E ( η ⊗2r t ⊗ η ⊗2r t-h ) ̸ = 0 m 4r ×1 ,
since the first elements of the two vectors

Y (2r) t ⊗ Y (2r) t-h and η ⊗2r t ⊗ η ⊗2r t-h are given by ( Y (2r) t ⊗ Y (2r) t-h ) 1 = exp {r (X t,1 + X t-h,1 )} and ( η ⊗2r t ⊗ η ⊗2r t-h ) 1 = η 2r t,1 η 2r t-h,1 ,
from which we can easily see that the first element of vector E

[ ε ⊗2r t ⊗ ε ⊗2r t-h ] ( E [ ε ⊗2r t ⊗ ε ⊗2r t-h ]) 1 = ( E [ Y (2r) t ⊗ Y (2r) t-h ]) 1 ( E [ η ⊗2r t ⊗ η ⊗2r t-h ]) 1 = E (exp {r (X t,1 + X t-h,1 )}) E ( η 2r t,1 ) E ( η 2r t-h,1 ) = E (exp {r (X t,1 + X t-h,1 )}) ( (2r)! 2 r r! ) 2 ( σ (t) η 1 σ (t-h) η 1 ) 2r > 0,
where

σ (t)
η 1 is the standard deviation of the first component of the random vector η t .

Estimation methodology

In this section, we propose two methodologies in order to estimate our model. The first one is the quasi-maximum likelihood method based on periodic Kalman filter. The second one uses the maximum likelihood and is based on the EM algorithm with particle filters and smoothers. We, more particularly, thoroughly describe the second one which turns out to be more useful in practice.

Quasi-maximum likelihood method based on periodic Kalman filter

In order to estimate the parameters of the multivariate non-periodic AR-SV model, [START_REF] Harvey | Multivariate stochastic variance models[END_REF] proposed a quasi-maximum likelihood (QM L) method. It is based on the state-space representation after setting the observed variable as a logarithm of the vector ε t ⊙ ε t , where the operator ⊙ is the Hadamard (or element-by-element) product. In our periodic case, model (3) can be linearized by taking the same transformation ln (ε s+τ S ⊙ ε s+τ S ) where we get the following state-space representation

{ Z s+τ S = X s+τ S +d s +u s+τ S , X s+τ S = α s + Φ s X s+τ S-1 + e s+τ S , τ ∈ Z, 1 ≤ s ≤ S, (6)
where

Z s+τ S = (Z s+τ S,1 , Z s+τ S,2 , ..., Z s+τ S,m ) ′ , Z s+τ S = ln (ε s+τ S ⊙ ε s+τ S ) , d s = E [ln (η s+τ S ⊙ η s+τ S )]
and

u s+τ S = ln (η s+τ S ⊙ η s+τ S ) -E [ln (η s+τ S ⊙ η s+τ S )] .
Let us remark that even though the random process η s+τ S is Gaussian, the measurement equation errors, u s+τ S , in (6) are nonnormal. The mean vector d s of ln (η s+τ S ⊙ η s+τ S ) and the (i, j)-th element of the covariance matrix Σ (s) u of u s+τ S can be given (see [START_REF] Harvey | Multivariate stochastic variance models[END_REF] respectively for s = 1, ..., S, by

d s = ( -1.2749 + log ( Σ (s) η (1, 1) ) , . . . , -1.2749 + log ( Σ (s) η (m, m) )) ′ ,
and

Σ (s) u (i, j) =      ∞ ∑ n=1 (n-1)! n( ∏ n k=1 ( 1 2 +k-1)) ( Σ (s) η (i,j) √ Σ (s) η (i,i) √ Σ (s) η (j,j) ) 2n , for i ̸ = j, i, j = 1, ..., m, π 2 2 , otherwise.
This last representation of the model makes the estimation problem more evident via QM L method as suggested by [START_REF] Harvey | Multivariate stochastic variance models[END_REF] for the multivariate non periodic case and [START_REF] Boussaha | On Periodic Autoregressive Stochastic Volatility Models[END_REF] for the univariate periodic case. However, the transformation of the model causes loss of information in the multivariate case. More specifically, an estimation could be made about the absolute values of the unknown parameters in Σ (s)

η , s = 1, ..., S, namely the Σ (s) η (i, j)'s, and the covariances between different η t,i 's, but their signs could not be estimated. This is due to the loss of information when the observations are squared. To solve this issue, [START_REF] Harvey | Multivariate stochastic variance models[END_REF] suggested to use the signs of the untransformed observations to obtain the sign of the covariance coefficients. They have proposed to estimate the sign of Σ (s) η (i, j) as positive if more than one-half of the pairs ε t,i ε t,j are positive.

To discuss the QM L method based on periodic Kalman filter, let Z s+τ S|s+τ S-1 be the best linear (one step ahead) predictor of Z s+τ S based on Z 1 , Z 2 , ..., Z s+τ S-1 , and let u s+τ S = Z s+τ S -Z s+τ S|s+τ S-1 be the sample innovation at time s + τ S, with mean square error Ω s+τ S = E ( u s+τ S u ′ s+τ S

) . The innovation at time s + τ S is defined as

u s+τ S = Z s+τ S -X s+τ S|s+τ S-1 -d s ,
where X s+τ S|s+τ S-1 is the best linear predictor of X s+τ S based on Z 1 , Z 2 , ..., Z s+τ S-1 , with mean square error

P s+τ S|s+τ S-1 = E [ ( X s+τ S -X s+τ S|s+τ S-1 ) ( X s+τ S -X s+τ S|s+τ S-1 ) ′ ] .
For a given realization ε = (ε 1 , ε 2 , ..., ε n ) generated from model (2), the quasi-likelihood of the parameter vector θ = (θ ′ 1 , ..., θ ′ S ) ′ , where

θ s = ( α s , (vech (Φ s )) ′ , ( vech
( Σ (s) e )) ′ , ( vech
( Σ (s) η )) ′ ) ′
, can be written in the innovation form as follows

L(θ; Z) = (2π) -nm 2 S ∏ s=1 τ 1 -1 ∏ τ =0 (det Ω s+τ S ) -1 2 exp { - 1 2 u ′ s+τ S Ω -1 s+τ S u s+τ S } , (7)
in which we need to evaluate Ω s+τ S and u s+τ S , for s = 1, ..., S and τ = 0, ..., τ 1 -1, where, for simplicity purposes, n can be taken as n = τ 1 S. Note that the sample innovation and its mean square error involved in ( 7) can be recursively computed using either the Kalman filter [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF] or the periodic Chandrasekhar-type recursions (Aknouche and Hamdi, 2007).

Clearly, from the state-space representation (6), the periodic Kalman filter, as it is well known, is given by the following recursions:

                 u s+τ S = Z s+τ S -X s+τ S|s+τ S-1 -d s , Ω s+τ S = P s+τ S|s+τ S-1 + Σ (s) u , K s+τ S = Φ s+1 P s+τ S|s+τ S-1 Ω -1 s+τ S , X s+τ S+1|s+τ S = α s+1 + Φ s+1 X s+τ S|s+τ S-1 + K s+τ S u s+τ S , P s+τ S+1|s+τ S = Σ (s+1) e +Φ s+1 ( P s+τ S|s+τ S-1 -P s+τ S|s+τ S-1 Ω -1 s+τ S P s+τ S|s+τ S-1 ) Φ ′ s+1 ,
with start-up values

X 1|0 = ( I m - S-1 ∏ v=0 Φ 1-v ) -1 S-1 ∑ v=0 ( v-1 ∏ i=0 Φ 1-i ) α 1-v ,
and

vec ( P 1|0 ) =   I m 2 - ( S-1 ∏ v=0 Φ 1-v ) ⊗2   -1 × S-1 ∑ v=0 ( v-1 ∏ i=0 Φ 1-i ) ⊗2 [ ( α 1-v -µ (1-v) X + Φ 1-v µ (S-v) X ) ⊗2 + vecΣ (1-v) e ] .
In practice, it is very difficult to obtain explicit formula of the QM L estimator, θ, which maximizes (7), with respect to θ. One thus needs to use numerical methods. It is also important to note that, under appropriate conditions, this QM L estimator, θ, has been shown to be consistent and asymptotically normally distributed (Ljung and Caines, 1979, p. 36). Despite these desirable asymptotic properties, the approximations provided by the periodic Kalman filter become less effective in our case where the normality hypothesis is abandoned. Indeed, the QM L estimator is not necessarily the best estimator for finite samples. For the periodic linear state-space models and when the sample size is small, [START_REF] Guerbyenne | Bootstrapping Periodic State-Space Models[END_REF] have provided numerous examples which indicate the poor finite sample properties of θ. To improve the finite sample performance of the QM L estimator, they suggested to use some procedures based on a bootstrap method to fit a periodic time series model expressed in state-space form. Therefore, as put forward, in the univariate SV model (see e.g. [START_REF] Kim | Fitting stochastic volatility models in the presence of irregular sampling via particle methods and the em algorithm[END_REF] for the non periodic case and Boussaha and Hamdi, 2015 for the periodic case), alternative methods to the traditional Kalman filter, such as the sequential Monte Carlo ones, should be considered.

Maximum Likelihood based on the EM algorithm with particle filters and smoothers

We now describe another estimation approach which combines the particle filters and smoothers with expectation-maximization (EM ) algorithm.

The particle filters are sequential Monte Carlo methods which can be applied to the general state-space models. These filters can be considered as a powerful alternative to the Kalman filter for the optimal estimation problems in a non-linear/non-Gaussian state-space framework (see e.g. [START_REF] Doucet | A tutorial on particle filtering and smoothing: Fifteen years later[END_REF]

. Let Y = (X ′ 0 , X ′ 1 , ..., X ′ n , ε ′ 1 , ε ′ 2 , ..., ε ′ n ) ′ and X = (X ′ 0 , X ′ 1 , .
.., X ′ n ) ′ denote the vector containing, respectively, the complete data and the log-volatilities data. For a given realization ε = (ε 1 , ε 2 , ..., ε n ) of model ( 2), the complete log-likelihood function of the parameter vector θ can be expressed as follows:

-2L(θ; Y ) = C + n ∑ t=1 log ( det ( Σ (t) e )) + n ∑ t=1 log ( det ( Σ (t) η )) + n ∑ t=1 m ∑ i=1 X t,i + n ∑ t=1 (X t -α t -Φ t X t-1 ) ′ ( Σ (t) e ) -1 (X t -α t -Φ t X t-1 ) + n ∑ t=1 ε ′ t diag ( exp { - 1 2 X t }) ( Σ (t) η ) -1 diag ( exp { - 1 2 X t }) ε t ,
where C is a constant independent of θ. It is well known that, as in a variety of cases where the data is considered to be incomplete, the estimation problem can be easily done via the simple recursive EM algorithm introduced by [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF]. Recall that this algorithm works in an iterative way.

In each iteration, there are two steps called E-step and M -step. To start the (i + 1)-th iteration, we have the parameter estimation from the last iteration (or initial value) θ (i) , and we define the following Q-function in the E-step

Q(θ, θ (i) ) = E [ -2L(θ; Y )|Y, θ (i) ] = C + n ∑ t=1 log ( det ( Σ (t) e )) + n ∑ t=1 log ( det ( Σ (t) η )) + n ∑ t=1 m ∑ i=1 X (n) t,i + n ∑ t=1 tr ( ( Σ (t) e ) -1 A t ) + n ∑ t=1 tr ( ( Σ (t) η ) -1 B t ) .
where

A t = X (n) t,t -X (n) t α ′ t -X (n) t,t-1 Φ ′ t -α t ( X (n) t )′ + α t α ′ t + α t ( X (n) t-1 )′ Φ ′ t -Φ t X (n) t-1,t + Φ t X (n) t-1 α ′ t + Φ t X (n) t-1,t-1 Φ ′ t , B t = E [ diag ( exp { - 1 2 X t }) ε t ε ′ t diag ( exp { - 1 2 X t }) Y, θ (i) ] , X (n) t = E ( X t |Y, θ (i)
) ,

and

X (n) t-h,t-k = E ( X (n) t-h X ′ t-k Y, θ (i)
)

, for h, k = 1, 0

Before going to the M -step, we need to evaluate the quantities X (n)

t , X (n)

t-h,t-k , A t and B t . As put forward, in the univariate AR-SV model (see [START_REF] Kim | Fitting stochastic volatility models in the presence of irregular sampling via particle methods and the em algorithm[END_REF][START_REF] Kim | Fitting stochastic volatility models in the presence of irregular sampling via particle methods and the em algorithm[END_REF][START_REF] Boussaha | On Periodic Autoregressive Stochastic Volatility Models[END_REF]Hamdi, 2016), these quantities can be sequentially approximated in time by using the particle filtering and smoothing algorithms. The following is the algorithm for the filtering step, from which we will obtain M samples from the probability density function p (X t |F t ), where F t denotes the σ-algebra based on the information available up to time t.

Algorithm 1 Particle filter algorithm

1: Initialization: sample from p 0 (X 0 ) to obtain f (j) 0 with initial weights w (j) 0 = 1/M , (for j = 1, ..., M ), where M is the number of the particles. 2: Iterate, for s = 1, ..., S, and τ = 0, ..., τ 1 -1. a. For j = 1, ..., M i. Simulate e (j)

s+τ S ∼ N (0, Σ (s) e ). ii. Compute p (j) s+τ S = α s + Φ s f (j) s+τ S-1 + e (j) s+τ S .
iii. Evaluate importance weights: compute

w (j) s+τ S = w (j) s+τ S-1 p ( ε s+τ S p (j) s+τ S ) ∝ w (j) s+τ S-1 × exp { - ( diag { exp ( -1 2 p (j) s+τ S
)} εs+τS

) ′ (Σ (s) η ) -1 ( diag { exp ( -1 2 p (j) s+τ S )} εs+τS ) 2 } × ( det ( Σ (s) η )) -1/2 det ( diag ( exp ( - 1 2 p (j) s+τ S
))) .

b. For j = 1, ..., M, normalize weights: compute

w (j) s+τ S = w (j) s+τ S / M ∑ j=1 w (j) s+τ S .
c. Compute the measure of degeneracy

n ef f = 1 / ∑ M j=1 w (j)2 s+τ S . If n ef f ≤ n T (typically n T = M/2) resample {( p (j) s+τ S , w (j) s+τ S ) , j = 1, ..., M } to obtain M equally-weighted particles {( f (j) s+τ S , 1/M ) , j = 1, ..., M } . 3: Finally, the sequence of M particles { f (j) s+τ S ; j = 1, ..., M
} is a random sample from p (X s+τ S |F s+τ S ) , for s = 1, ..., S and τ = 0, ..., τ 1 -1.

Algorithm 2 Particle smoothing algorithm

1: For j = 1, ..., M, choose s (j) n = f (i) n , with probability w (i) n . Let us fix W (j) n = 1/M. 2: For j = 1, ..., M.
a. For s = S, ..., 1 and τ = τ 1 -1, ..., 0, calculate for i = 1, ..., M

W (i) s+τ S-1|τ S+s = w (i) s+τ S-1 × exp { - ( s (j) s+τ S -αs-Φsf (i) s+τ S-1 ) ′ ( Σ (s) e
) -1 ( s

(j) s+τ S -αs-Φsf (i) s+τ S-1 ) 2 } × ( det ( Σ (s) e
)) -1/2 .

b. For i = 1, ..., M, normalize the smoothed weights via

W (i) s+τ S-1|s+τ S = W (i) s+τ S-1|s+τ S / M ∑ j=1 W (i) s+τ S-1|s+τ S . c. Draw s (j) s+τ S-1 from f (i) s+τ S-1 , with probability proportional to { W (i) s+τ S-1|s+τ S , i = 1, ..., M } .
3: Finally, compute s = 1, ..., S and τ = 0, ..., τ 1 -1,

X (n) s+τ S = ∑ M j=1 s (j) s+τ S M , P (n) s+τ S = ∑ M j=1 ( s (j) s+τ S -X (n) s+τ S ) ( s (j) s+τ S -X (n) s+τ S ) ′ M -1 , P (n) s+τ S,s+τ S-1 = ∑ M j=1 ( s (j) s+τ S -X (n) s+τ S ) ( s (j) s+τ S-1 -X (n) s+τ S-1 ) ′ M ,
and

B s+τ S = ∑ M j=1 diag ( exp { - s (j) s+τ S 2 }) ε s+τ S ε ′ s+τ S diag ( exp { - s (j) s+τ S 2 }) M .
After replacing

X (n) t , P (n) t , P (n)
t,t-1 , A t and B t with their approximations, we thus go to the M -step, where the estimated parameter is obtained as follows

θ (i+1) = arg max θ Q ( θ, θ (i)
) .

The first derivatives of the Q-function with respect to θ are as follows

∂Q(θ, θ (i) ) ∂α s = ( Σ (s) e ) -1 τ 1 -1 ∑ τ =0 ( 2α s -2X (n) s+τ S + 2Φ s X (n) s+τ S-1
) , ( 8)

∂Q(θ, θ (i) ∂ (vech (Φ s )) = vech ( ∂Q(θ, θ (i) ) ∂Φ s ) , ( 9
)
∂Q(θ, θ (i) ) ∂ ( vech ( Σ (s) e )) = D ′ m × vec ( ∂Q(θ, θ (i) ) ∂Σ (s) e ) , ( 10
) ∂Q(θ, θ (i) ) ∂ ( vech ( Σ (s) η )) = D ′ m × vec ( ∂Q(θ, θ (i) ) ∂Σ (s) η ) , ( 11
)
where

∂Q(θ, θ (i) ) ∂Φ s = 2 ( Σ (s) e ) -1 × τ 1 -1 ∑ τ =0 [ α s ( X (n) s+τ S-1 )′ - ( X (n) s+τ S ( X (n) s+τ S-1 )′ + P (n) s+τ S,s+τ S-1 ) +Φ s ( X (n) s+τ S-1 ( X (n) s+τ S-1 )′ + P (n) s+τ S-1 )] , ∂Q(θ, θ (i) ) ∂Σ (s) e = ( Σ (s) e ) -1 τ 1 -1 ∑ τ =0 [ I m -A s+τ S ( Σ (s) e ) -1 ] , ∂Q(θ, θ (i) ) ∂Σ (s) η = ( Σ (s) η ) -1 τ 1 -1 ∑ τ =0 [ I m -B s+τ S ( Σ (s) η ) -1 ] ,
and where D m denotes the

( m 2 × 1 2 m (m + 1)
) duplication matrix such that vec (A) = D m × vech (A), for any (m × m)-matrix A (see e.g. Lütkepohl, 1996, Section 9.5). Here vech is the half column staking operator.

The estimates of the parameters θ (i+1) can then be obtained by equating ( 8), ( 9), ( 10) and ( 11) to zero. Hence, at the (i + 1)-th iteration, the parameter estimates of α s , vech (Φ s ), vech

( Σ (s) e )
and vech

( Σ (s) η ) , for s = 1, ..., S, are summarized below                                        vech ( Σ (s) η ) = 1 τ 1 ∑ τ 1 -1 τ =0 vech ( B s+τ S
) , vech

( Φ s ) = [ L m {( 1 τ 1 ( ∑ τ 1 -1 τ =0 X (n) s+τ S-1 ) ( ∑ τ 1 -1 τ =0 X (n) s+τ S-1 ) ′ - ∑ τ 1 -1 τ =0 ( X (n) s+τ S-1 ( X (n) s+τ S-1 )′ + P (n) s+τ S-1
))

⊗ I m } L ′ m ] -1 ×vech ( 1 τ 1 ( ∑ τ 1 -1 τ =0 X (n) s+τ S ) ( ∑ τ 1 -1 τ =0 X (n) s+τ S-1 ) ′ - ∑ τ 1 -1 τ =0 ( X (n) s+τ S ( X (n) s+τ S-1 )′ + P (n) s+τ S,s+τ S-1
)) ,

α s = 1 τ 1 ∑ τ 1 -1 τ =0 X (n) s+τ S -1 τ 1 Φ s ∑ τ 1 -1 τ =0 X (n) s+τ S-1 , vech ( Σ (s) e ) = 1 τ 1 ∑ τ 1 -1 τ =0 vech ( A s+τ S
) ,

where L m denotes the

( 1 2 m (m + 1) × m 2 ) elimination matrix such that vech (A) = L m × vec (A) ,
for any (m × m)-matrix A (see e.g. Lütkepohl, 1996, Section 9.6).

Application

This final section provides an empirical application. More precisely, we first describe our data and provide some preliminary analysis in order to justify the use of periodic and multivariate models. We thus apply the theoretical analysis developed in Section 2 to provide a multivariate modeling of oil spot price and three exchange rates time series.

Data and preliminary analysis

We model four time series: the daily Saharan Blend oil (SB) spot price time series (provided by the Algerian Ministry of Energy) and three daily time series of exchange rates: Euro/U.S. dollar (EU R/U SD) provided by Thomson Reuters, Euro/Algerian dinar (EU R/DZD) and U.S. dollar/ Algerian dinar (U SD/DZD) provided by the Bank of Algeria. We first removed all the days in which the SB market was closed. The period of analysis is 3 January 2005 to 31 December 2015. The final sample consists of 2836 observations for each series.

In this empirical study, the model is structured as follows. For notational purposes, we collect the original Saharan Blend oil price and three exchange rates at time t in the vector

Y t =     Y t,1 Y t,2 Y t,3 Y t,4     =     SB t EU R/U SD t EU R/DZD t U SD/DZD t     ,
and their corresponding log-return series in the vector

Z t = (Z t,1 , Z t,2 , Z t,3 , Z t,4 ) ′ , where Z t,i = log (Y t,i ) -log (Y t-1,i ) , for i = 1, 2, ..., 4.
The original series Y t,i and the log-return series Z t,i are respectively shown in Figures 1 and2. It turns out that all the original series Y t,i are nonstationary while the log-return series appear to be stationary. Moreover, the volatility clustering phenomenon is clearly evident in each of them. Not surprisingly, the volatility was higher during the crisis in 2008 and after 2015. The volatilities in the log-returns series are subject to some breaks: high volatility between April 2008 and July 2009 and after 2015 and low volatility between December 2006 and March 2008 and between August 2012 and December 2014. Additionally, in the whole study period the SB log-return is the most volatile, followed by the EU R/U SD and EU R/DZD log-returns. The U SD/DZD log-return is the most stable one. Some descriptive statistics are provided for the four log-return series in Tables 1234. These statistics are very similar across these series. Their standard deviations are much greater than the means in absolute value, indicating that the means are not significantly different from zero. Moreover, their excess kurtosis is significantly positive. Therefore, these series have heavy tails relative to the normal distribution, which is also typical in these financial data. The Jarque-Bera test rejects the normality of the four log-return distributions. Table 5 provides the sample correlation matrix for the entire study period. Correlation coefficients vary from -0.5591 to 0.6720. The strongest negative correlation is for the pair (EU R/U SD, U SD/DZD) and the strongest positive correlation is for the pair (EU R/U SD, EU R/DZD). The SB is the least correlated with all the exchange rates. This suggests that estimating a joint model may yield interesting information on the relationships between our data series. This is why we do use, in the sequel, a multivariate modelling of the collected data. Tables 1-4 also report the descriptive statistics of the four log-returns for each day of the week. The findings indicate that for the U SD/DZD (resp. EU R/DZD, EU R/U SD, SB), the lowest (-0.0132) (resp. -0.0233, -0.0332, -0.1244) and the highest (0.0335) (resp. 0.0497, 0.0381, 0.1367) returns are observed in Friday (resp. Friday and Thursday, Monday and Thursday, Friday and Monday). For the analyzed period, descriptive statistics per day register the lowest standard deviation of 0.0023 (resp. 0.0041, 0.0058, 0.0182) in Tuesday (resp. Tuesday, Tuesday, Wednesday) for the U SD/DZD log-returns series (resp. EU R/DZD, EU R/U SD, SB), while the highest standard deviation of 0.0033 (resp. 0.0049, 0.0072, 0.0212) is observed for Friday (resp. Monday and Thursday, Monday, Monday). The skewness for all the days of the week and for all log-return series is positive, while it is negative for Monday and Tuesday in EU R/DZD, for Monday in EU R/U SD and for Tuesday, Wednesday and Friday in SB log-returns. Furthermore, one of the features which prominently stands out most from Tables 1234is that the kurtosis for each day-of-the-week is much larger than 3. This reflects the fact that the tails of the distributions of all the analyzed log-returns per day are fatter than the tails of the normal distribution. These descriptive statistics of the four log-returns for each day of the week are not surprising since it is usually observed in financial time series where correlations between daily returns and volatility display some day-ofthe-week effects (see e.g. [START_REF] Franses | Modelling day-of-the-week seasonality in the S&P 500 index[END_REF][START_REF] Hamdi | Mixture Periodic GARCH Models: Theory and Applications[END_REF]. Such a property may be useful as sometimes one may expect economic agents to have different behavior in different days of the week. Indeed, from daily first-order autocorrelations for our return series (see Table 6), it is evident that returns on Mondays are positively correlated with those on the preceding Fridays, while returns on Tuesdays are negatively correlated with those on Mondays. This is why we do consider the periodicity as an important issue in our modelling of these time series.

Empirical results

In the following, we propose a periodic model that allows for the description of the day-of-the-week or intraweek effect in the daily exchange rate and oil price series. Such a formulation with period S = 5 seems to be well adapted to explain the presence (or not) of intraweek seasonality as have done [START_REF] Franses | Modelling day-of-the-week seasonality in the S&P 500 index[END_REF] for modelling day-of-the-week seasonality in the S&P 500 index through a univariate periodic AR-GARCH process. We will consider the 5-periodic P V AR-SV model, where the parameters are allowed to vary with the day of the week (s = 1, 2, 3, 4 and 5)

s =         
1 if the day corresponding is a Monday, 2 if the day corresponding is a Tuesday, . . . 5 if the day corresponding is a Friday.

From Table 6, it is clear that the series Z t , t = 1, ..., 2835, exhibits a periodicity in the conditional mean. In order to capture such a periodicity, which is not possible by the use of a pure P V AR-SV model, we have applied a multivariate periodic autoregressive (P V AR) filter to Z t and we have considered the residuals as the underlying series for our P V AR-SV modeling. Table 7 shows the results of the application of the filter.

The plot of the real series Z t,i , i = 1, ..., 4, the adjusted and the residuals ones computed from the P V AR (2) fitted model are given by Figure 3. As a second step, we use these residuals to estimate the volatility.

The estimated parameters of the 5-periodic P V AR-SV model are presented in Table 8. We observe that all the log-return series examined exhibit highly persistent volatilities. The persistent coefficient ranges from ∏ S s=1 ϕ (s)

i,i = 0.7960 for U SD/DZD to 0.7543 for SB. This allows us to conclude that the P V AR-SV estimated model is periodically stationary as the quantity characterizing the estimated processes, namely ρ ( ∏ S s=1 Φ s ) defined in ( 5) , is smaller than unity (ρ ( ∏ S s=1 Φ s ) = 0.7960). On the other hand, when we analyze the estimated model period by period, we notice that only the models corresponding to Wednesday and Thursday are nonstationary. Indeed, we have ρ (Φ s ) equal, respectively, to 0.9971, 0.9571, 1.5110, 1.1020 and 0.9340 for s = 1, 2, 3, 4 and 5.

These results are consistent with the fact that exchange rates and oil price volatilities are often clustered. In all cases, the transmission coefficients (ϕ (s) i,j ) are quite weak and therefore there is a volatility interaction in daily data. For instance, from the estimated model, information in oil market is quickly transmitted and incorporated into the EU R/U SD, EU R/DZD and U SD/DZD exchange rates. This could suggest that volatility in the oil market have significant impacts on the FX market, especially on Algerian dinar.

Another interesting phenomenon appears in our empirical analysis. It deals with the fact that the seasonal return shock correlation

Σ (s) η (i,j) √ Σ (s) η (i,i) √ Σ (s) η (j,j)
(s = 1, ..., 5 and i, j = 1, ..., 4) is significantly negative between U SD/DZD and the other series, while it is significantly positive in all the other cases (between EU R/U SD and SB and between EU R/DZD and SB and EU R/U SD).

Figure 5 shows the estimated volatilities computed from 5-periodic P V AR-SV model of SB, EU R/U SD, EU R/DZD and U SD/DZD. We can observe that the estimated volatilities reflect quite well the variation of the residuals. Furthermore, from Table 9 it turns out that the empirical coverages of the P V AR-SV -based prediction intervals are closer to the nominal coverages. These results show that the P V AR-SV model, fitted to the residuals computed from P V AR modeling of our daily log-return time series, seems well accurate and has a well volatility forecasting performance.

: Z s+5τ = µ s + Ψ s,1 Z s+5τ -1 + Ψ s,2 Z s+5τ -2 + ε s+5τ . µ s Ψ s,1 Ψ s,2 s = 1     0 0 0 0.0003         0

Figure 1 :

 1 Figure 1: The original series.

Figure 5 :

 5 Figure 5: Estimated volatility from P V AR-SV 5 model.

Table 1 .

 1 Descriptive statistics of SB log-returns for each day of the week as well as the entire study period

		Monday Tuesday Wednesday Thursday Friday	All days
	# of obs.	567	570	572	569	557	2835
	Mean	-0.0014	0.0000	0.0005	0.0011	-0.0004 -2.29 × 10 -5
	Median	0.0000	0.0000	0.0011	0.0020	-0.0003	9.32 × 10 -5
	Maximum	0.1367	0.0633	0.0850	0.1067	0.0825	0.1367
	Minimum	-0.0777 -0.1119	-0.0812	-0.0909	-0.1244	-0.1244
	Std. Dev.	0.0212	0.0199	0.0182	0.0211	0.0193	0.0110
	Skewness	0.7466	-0.4172	-0.0594	0.0078	-0.3224	0.0276
	Kurtosis	8.5317	5.4482	5.0156	5.9985	7.9584	6.7408
	Jarque-Bera 775.5827 158.8808	97.1668	213.1664 580.2407	1653.3850

Table 2 .

 2 Descriptive statistics of EU R/U SD log-returns for each day of the week as well as the entire study period

		Monday Tuesday Wednesday Thursday Friday	All days
	# of obs.	567	570	572	569	557	2835
	Mean	-0.0005 -0.0003	0.0000	0.0002	0.0001	-7.56 × 10 -5
	Median	-0.0001 -0.0001	0.0000	0.0001	0.0002	0.0000
	Maximum	0.0236	0.0264	0.0272	0.0381	0.0301	0.0381
	Minimum	-0.0332 -0.0213	-0.0206	-0.0254 -0.0212 -0.0332
	Std. Dev.	0.0072	0.0058	0.0063	0.0065	0.0067	0.0065
	Skewness	-0.2244	0.2663	0.1105	0.2828	0.1631	0.0833
	Kurtosis	4.3099	5.6897	4.5673	5.9422	4.5940	5.0206
	Jarque-Bera 45.2972 178.5543	59.7092	212.8093	61.4376 485.5663
	Table 3. Descriptive statistics of EU R/DZD log-returns for each day of the week as well as
	the entire study period
		Monday Tuesday Wednesday Thursday	Friday	All days
	# of obs.	567	570	572	569	557	2835
	Mean	-0.0002 -0.0002	0.0003	0.0003	0.0001	6.32 × 10 -5
	Median	-0.0001	0.0001	0.0002	0.0004	0.0000	0.0001
	Maximum	0.0209	0.0170	0.0313	0.0497	0.0226	0.0497
	Minimum	-0.0233 -0.0230	-0.0158	-0.0195	-0.0220 -0.0233
	Std. Dev.	0.0049	0.0041	0.0044	0.0049	0.0048	0.0046
	Skewness	-0.0819 -0.4649	0.8022	2.2173	0.1346	0.6007
	Kurtosis	6.2042	6.6266	8.9065	25.1483	6.8762	11.6985
	Jarque-Bera 243.1895 332.8976	892.8278	12096.3000 350.3907 9108.4040

Table 4 .

 4 Descriptive statistics of U SD/DZD log-returns for each day of the week as well as

	the entire study period
		Monday	Tuesday Wednesday Thursday Friday	All days
	# of obs.	567	570	572	569	557	2835
	Mean	0.0003	0.0000	0.0000	0.0001	0.0003	0.0001
	Median	0.0001	0.0000	-0.0001	-0.0001	0.0001	-1.89 × 10 -5
	Maximum	0.0282	0.0094	0.0138	0.0181	0.0335	0.0335
	Minimum	-0.0092	-0.0113	-0.0070	-0.0100	-0.0132	-0.0132
	Std. Dev.	0.0030	0.0023	0.0024	0.0028	0.0033	0.0028
	Skewness	1.8854	0.0974	1.0514	1.0981	2.8623	1.8005
	Kurtosis	17.9821	6.3905	8.5105	9.5237	27.1108	18.9315
	Jarque-Bera 5638.8570 273.9248	829.1143	1123.3530 14252.2300	31513.4400

Table 5

 5

	. Correlation matrix of log returns
	log-returns	SB EU R/U SD EU R/DZD U SD/DZD
	SB	1	0.0491	0.0926	-0.1260
	EU R/U SD		1	0.6720	-0.5591
	EU R/DZD			1	-0.4003
	U SD/DZD				1

Table 6

 6

	. Periodic autocorrelations of log-return series
	log-returns	Monday Tuesday Wednesday Thursday Friday
	SB	0.0565	-0.0383	0.0252	0.0596	0.0241
	EU R/U SD	0.0272	-0.1629	0.0182	-0.0006 -0.0339
	EU R/DZD	0.0424	-0.0158	0.0231	0.2094	0.0877
	U SD/DZD	0.1617	-0.0617	-0.0685	0.1639	0.1728

Table 7 .

 7 P V AR filter to Z

	t

Table 8 .

 8 Parameter estimates of 5-periodic P V AR-SV model for ε

	1/2 s+5τ η s+5τ ,
	s+5τ = V
	s+5τ : ε

s + Φ s X s+5τ -1 + e s+5τ , V t = diag (exp X t ) , η

Table 9 .

 9 Empirical coverages of the (1 -α)100% one-step-ahead prediction intervals .62 70.37 79.41 87.71 92.02 96.36 U.S. dollar/Algerian dinar 53.50 62.61 71.68 79.31 87.25 91.56 95.97

			(1 -α)100%
		50%	60%	70%	80%	90%	95%	99%
	Saharan Blend oil prices	53.81 62.50 71.61 79.80 87.96 92.69 96.96
	Euro/U.S. dollar	53.25 62.96 72.60 81.04 89.65 93.33 97.78
	Euro/Algerian dinar	52.37 61


Balázs Égert
email: balazs.egert@oecd.org.

Alain De Serres

Yvan Gillemette

Alberto González Pandiella

Mikkel Hermansen

Vincent Koen

Jan Strasky

David Turner

Ben Westmore

AGGREGATE MULTI-FACTOR PRODUCTIVITY: MEASUREMENT ISSUES IN OECD COUNTRIES

Keywords: multi-factor productivity, measurement, human capital, OECD AGGREGATE multi-factor productivity: measurement issues in OECD countries C23, C51, J2, L43, L51, O4

This paper analyses for 34 OECD countries the extent to which the calculation of aggregate multi-factor productivity (MFP) is sensitive to alternative parameterisations. The starting point is the definition of MFP used in previous work in the OECD's Economics Department (e.g. Johansson et al. 2013). They include alternative MFP measures, with human capital included or excluded, with different measures of Purchasing Power Parity (PPP) exchange rates, using time-varying capital depreciation rates and different measures of capital stock and labour input (headcount against hours worked). The main result of the paper is that whether or not human capital is included in MFP makes a significant difference for the level and dynamics of MFP. At the same time, MFP measures are less sensitive to other parameters of the calculation.

Introduction

1.

Measuring multi-factor productivity correctly at the macroeconomic level and understanding its main drivers is key to implementing pro-growth reforms. Progress has been made in measuring multi-factor productivity (MFP) and understanding its determinants at firm and industry level (O'Mahony and Timmer, 2009; Syverson, 2011; Andrews and Cingalo, 2014; Bartelsman et al., 2014). However, measuring MFP at the aggregate/macroeconomic level remains surrounded by measurement problems (Madsen 2006, 2014). Considerable work has been done in recent years to measure MFP but MFP is to some extent a measure of what we do not know. For instance, measures of MFP capturing the contribution of human capital within MFP may distort spatial and temporal comparisons. Indeed, the uncertainty about the precise level, comparability across countries, and the evolution over time has important implications for how precisely one is able to pin down the drivers of MFP in an empirical analysis. Not measuring accurately the MFP frontier masks our ability to reveal how far countries or firms are from the frontier and may lead to flawed conclusions regarding the speed of convergence to it. A wide margin of error may produce dubious results regarding the role of various institutions and policies for the evolution of MFP.

2.

Against this background, this paper proposes to carry out an assessment of MFP measurement using the approach adopted in earlier OECD Economics Department work (Johansson, 2013). It is important to note that the objective is not to create new OECD productivity measures but just to carry out an extensive sensitivity analysis. 2 The paper shows the differences for alternative MFP measures, with human capital included or excluded, with different measures of Purchasing Power Parity (PPP) exchange rates, using time-varying capital depreciation rates and different measures of capital stock and labour input (headcount against hours worked).

3.

The main results of this sensitivity analysis can be summarised as follows. First, whether or not human capital is included in MFP makes a significant impact for the level and dynamics of MFP. 3 The absolute level and also the time profile of MFP series, from which the effect of human capital are stripped off, are often counterintuitive. This might indicate measurement problems with human capital. Southern European countries such as Spain and Italy have been top performers in the 1980s and early 1990s in terms of MFP levels. The United States was in the middle of the distribution for quite some time. At the same time, Italy and Spain experienced a trend decline in their MFP over three decades. By contrast, MFP series that include the effect of human capital show more plausible patterns. Second, while time-varying Purchasing Power Parity (PPP) has a number of advantages over a constant PPP rate, MFP measures based on constant (2005) PPPs provide a more plausible ranking of OECD countries in terms of relative MFP levels. Whether employment or total hours worked are used for labour input or how exactly the capital stock is calculated do not make too much difference. This finding is not in line with other studies such as OECD (2017).

2.

Official OECD data for MFP are reported in the OECD Compendium of Productivity Indicators 2017 and available in the OECD productivity database.

3.

Whether MFP is obtained as output minus human and physical capital and labour (MFP excluding human capital) or as output minus physical capital and labour (MFP including human capital)

4.

The rest of the paper is organised as follows. Section 2 gives an overview on the calculation of MFP. Section 3 discusses the role of human capital in MFP calculations. Section 4 analyses the use of purchasing power parity (PPP) in the context of cross-country comparison. Sections 5 and 6 present MFP measures based on alternative labour and capital input definitions. Section 7 sketches out a framework for how MFP series should be chosen for empirical analysis from the large number of alternative MFP series produced in this paper. Section 8 finally provides some concluding remarks.

Calculating multi-factor productivity: a production function framework

5.

MFP can be calculated as a residual of output once all inputs including capital and labour are accounted for (see e.g. Johansson et al., 2013). Typically, the log-level of MFP is derived on the basis of equation ( 1) below. ln (MFP 𝑡 ) = ln(𝑌 𝑡 ) /𝜏 -ln(𝐻𝐾 𝑡 ) -ln(𝐿 𝑡 ) -(1 -𝜏)/𝜏 × ln (𝐾 𝑡 )

(1)

where 𝑌 𝑡 is real output, 𝐻𝐾 𝑡 , 𝐾 𝑡 and 𝐿 𝑡 are the stock of human and physical capital and labour input, respectively. 4 𝜏 is the share of labour and 1 -𝜏 the share of capital in the production function. 5 Y expressed in national currency units is sufficient to study the evolution of MFP over time. Cross-country comparison of MFP levels necessitates the conversion of output into a common currency unit. It is usually done using the purchasing power parity exchange rate (the ratio of absolute price levels in the domestic and foreign (reference) country), which accounts for different levels of economic development. Our benchmark MFP calculation is based on 2005 PPPs. 6 Labour force surveys typically consider people living in the country and therefore exclude people working but not living in the country 7 Most studies and the OECD's Productivity database are unaffected by this as the data on labour input is drawn from business surveys that measure input based on the 'domestic' as opposed to 'national' concept. However, this study uses labour force survey data and as a consequence an adjustment (CLF) is necessary as shown below. Equation ( 1) is therefore complemented by a labour force adjustment term (CLF):

ln (MFP 𝑡 ) = ln(𝑌 𝑡 ) /𝜏 -ln(𝐻𝐾 𝑡 ) -ln(𝐿 𝑡 ) -𝑙𝑛(𝐶𝐿𝐹 𝑡 ) -(1 -𝜏)/𝜏 × ln (𝐾 𝑡 ) (2)
The inclusion of human capital is related to the standard Jorgensonian approach towards quality-adjusting labour input. Human capital is approximated by the Mincer equation using mean years of schooling of total population (MYS). Throughout this paper, the Mincer equation will be parameterised in line with Johansson et al. (2013) as follows:

4. Equation (1) gives MFP which is consistent with Harrod-neutral labour augmenting technical progress. This paper does not address how alternative function forms of the production function would impact on the measure of MFP.

5.

Capital and labour shares can differ across countries and can change over time. Applying different factor shares across countries would make the comparison of productivity levels difficult. Applying factor shares changes over time would change the slope of the series, again making cross-country level comparison difficult. Indeed, factor shares are kept constant over time and the same factor shares are used for specific sectors across countries in firm-level productivity analysis (Gal, 2013).

6.

The absolute price level of a given country in 2005, divided by the absolute price level in the USA in 2005.

7.

For instance many French people work in Luxembourg and Switzerland but live in France.

𝐻𝐾 𝑡 = 0.1254 × 𝑀𝑌𝑆 -0.002 × 𝑀𝑌𝑆 2

(3) 6.

Figure 1 plots MFP series obtained from Johansson et al. (2013), labeffsd). These series are calculated using equations ( 2) and (3) with 𝜏 = 0.67, 2005 PPPs and the number of total employment. The underlying series are smoothed to filter out short-term disturbances. Figure 1 also shows our reference MFP series, calculated along the lines of equation (2) using raw (unsmoothed) data series for human and physical capital and labour (see Table 1). 8 The two sets of data have a number of common features:

 Some Southern European countries including Spain, Italy and, to a lesser extent, Portugal were among the highest productivity countries from the mid-1980s to the mid-1990s. 9

 Luxembourg, Ireland and Norway have been among the top performers over the last decade.

 The United States was in the middle of the pack in the 1980s and became a top performer after the 2007 crisis.

 Japan has been at the bottom of the productivity distribution. Its absolute productivity level is very close to those one can observe in Central and Eastern European countries.

 The absolute MFP level of Germany and Switzerland has been persistently in the low range of advanced OECD countries.

 A number of countries experienced a trend decline in MFP levels, especially for smoothed MFP series. Spain and Italy, which were among the highest-MFP countries in the 1980s, underwent decades of steady decay.

7.

These observations, especially the counterintuitive relative levels and the trend decline observed in some of the countries prompt questions about the strength of assumptions used to estimate MFP, and human capital. Against this backdrop, a systematic assessment of factors causing these counterintuitive results is carried out. At the centre of this exercise, a reference measure (MFP12) is constructed, which comprises the following elements:  Subtracting human capital from output (MFP obtained as output minus human and physical capital and labour)

 Using total employment (rather than hours worked)

 Capital stock with time-varying depreciation rates (drawn from the Penn World Table 8.0)

 2005 GDP PPP for output and 2005 investment PPP for the capital stock (from PWT 8.0).

8.

Recent studies using similar MFP series include Johansson et al., (2013), Westmore (2013) and Andrews and Westmore (2014).

9.

Not shown here, but Brazil and Saudi Arabia had very high MFP levels in the late 2000s.

Figure 1. MFP in selected OECD countries, 1981-2013

Note: LABEFFSD is the MFP series obtained from the OECD Economic Outlook database. MFP12 is the reference MFP series for further investigation. It is based on raw (unsmoothed) series (see Table 1).

MFP12 is very similar to the series used in Johansson et al (2013) and earlier vintages of OECD databases. Using MFP12 as a starting point, a number of alternative MFP measures are calculated to investigate the sensitivity of MFP measures to changes in the way they are calculated including i) the treatment of human capital, ii) whether the number of employment or total hours worked is used, iii) how the capital stock is calculated, and iv) the type of PPPs used to obtain MFP levels comparable across countries (see Table 1 for an overview). Overall, the treatment of human capital is what matters most. The type of PPP applied can also make a big difference. By contrast, the way the capital stock is constructed and the measurement of employment matters to a considerably lesser extent (Appendix B presents the alternative MFP measures for all countries). 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1

Human capital

8.

Whether or not human capital is included in MFP measures makes a significant difference. Human capital is measured on the basis of mean years of schooling. Mean years of schooling may not be strictly comparable across countries if education systems equip pupils with different knowledge and capabilities. Not adjusting for differences in the quality of teaching may seriously bias cross-country comparisons of human capital. 10 The PISA scores published by the OECD can be taken as a rough approximation for the quality of teaching, at a given point in time, up to the years of age 15. The mean years of schooling fairly represents the quantity of schooling. Plotting these two series against each other show very limited correlation (Figure 2). In particular, data are clustered: for instance, the mean years of schooling is about 13.5 years with a 10% variation in PISA scores. A similar effect can be observed when mean years of schooling is about 11.5 years.

10.

The proxy proposed by Madsen (2014) to adjust for the quality of teaching, the teacher-pupil ratio modified for weekly hours, appears to be problematic because it most probably do not account for quality differences in tertiary education.

Figure 2. The relationship between the quality and quantity of schooling

Note: the vertical axis depicts the overall PISA score 2012, the horizontal axis represents mean years of schooling (MYS) in the same year.

Source: OECD.

9.

Human capital based only on quantity of inputs and not adjusted for quality can have a substantial impact on the level and the dynamics of the MFP series. If human capital is subtracted from output (as in MFP12), changing all the other parameters does not alter the overall picture. In contrast, keeping human capital in MFP (MFP22) changes the overall profile of the series. The United States is among the top performers from 1985 to 2013. Italy and Spain have more plausible MFP levels and the decline only starts in the mid-1990s for Italy (Figure 3). It is important to recognise that MFP measures (such as MFP22) including human capital are consistent with MFP measures calculated at the sectoral and firm levels: it is indeed very difficult to measure sector-or firm-specific human capital (Gal, 2013). Constant PPPs are an accurate conversion factor (exchange rate) for temporal comparisons across countries if the real exchange rate is essentially unchanged, that is if the gap between PPPs and actual exchange rates remains stable over time and inflation rates are similar. In such a case, using the PPP rate rather than the market exchange rate would increase the productivity level of the less developed country, and hence lower the gap to the benchmark country. 11,12 But for emerging market and developing economies in particular, typically, the faster a country is catching up vis-à-vis advanced economies the more likely the use of constant PPPs will give rise to a bias in the measurement of TFP levels. The relative price level (or the level real exchange rate) of catching-up countries tends to increase compared to the benchmark countries (the United States in our case). A major explanation is the Balassa-Samuelson effect, 13 but other factors can also generate higher prices in catching-up economies. 14

11.

The PPP rate is given as the absolute price level of the domestic country expressed in national currency (P) divided by the absolute price level of the benchmark country (P*). That is, PPP=P/P*. The relative price level is the comparison of the two countries' price levels in the same currency. The domestic country's price level expressed in the benchmark country's currency is P/NER where NER is the exchange rate calculated as national currency units (NCU) over one unit of the foreign currency (FCU) NCU/FCU e.g. GBR/USD. The relative price level is (P/NER)/P*. This indicates whether the domestic country's absolute price level is lower or higher compared to the benchmark country. The relative price level can be rewritten as: (P/P*)/NER: the PPP rate divided by the nominal exchange rate. This is indeed the level real exchange rate.

12. The market exchange rate would underestimate the level of MFP in the less developed country, similarly as in the case of per capita income. The reason is that if that the relative price level of less developed countries is lower, P/P* < NER. 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1

The dynamic version

1 USA NOR CHE PRT ESP ITA JPN IRL TUR mfp22 Box 1.

The concepts of different PPP measures

Using 2005 PPP would, for the period prior to 2005, underestimate the level and overestimate the change in MFP of catching-up economies, as it implies a higher absolute domestic price level (P) and a higher conversion (exchange rate) rate (hence a smaller adjustment compared to the market exchange rate).

Using time-varying PPPs, which reflect real convergence phenomena, would potentially attenuate these biases. The so-called chain-linked PPP is one avenue. Chain-linked PPP is constructed by connecting different vintages of absolute price levels through some kind of interpolation.

15 Nevertheless, chained PPPs are not without problems. Most importantly, comparing different vintages of PPPs can be very problematic. Generally, it is not recommend the use of GDP numbers converted using time-varying PPPs from different vintages for time series analysis (Eurostat-OECD (2016), and Feenstra et al., 2013, Table 5). 16 Nevertheless, a potential problem is that current PPPs are not consistent with GDP deflators.

Chain-linked PPPs are subject to quality and new goods biases. Ideally, PPPs should measure the price of almost identical goods. The International Comparison Programme, which is the basis of PPPs, notes that goods and services included in a country's price level is mostly based on a broad specification, precisely because the same good may not be available or representative in a country. This means that more developed countries may have higherquality and hence higher-priced goods in their price basket. As a result, PPPs may overestimate cross-country price differences. But this has also important implications for the time series dimension. With real catching-up, there will be a shift from lower quality to higher quality goods and services and the variety of goods can increase (new goods can enter and old items exit the basket). Higher quality and new goods show up in higher price levels from one price survey to another one. This potentially overstates price level convergence.

Quality and new-goods bias are usually taken care of in consumer price indices, even if imperfectly. This argues for the use of a single PPP vintage (for instance 2005 PPPs) but which are extended backward and forward in time by cumulated inflation differentials (the evolution of the level CPI in the domestic country relative to the one in the benchmark countrythe USA in our case). The main caveat is, however, that inflation differentials tend to largely underestimate real convergence compared to price level convergence. Weights used in the CPI basket represent the consumption basket of the average household in any given country. According to Engel's law, poorer households spend a larger proportion of their income on food than on other items. Analogously, they also spend less on services. In fact, the share of services in the CPI basket of developing/emerging market economies tend to be considerably lower than in more advanced countries. The low weight of services will considerably understate the Balassa-Samuelson effect in consumer price indices. A mid-way would be to rely on GDP deflators. The weight of the services sector therein is considerably higher than in the CPI basket. The Balassa-Samuelson effect will not be hidden in the GDP deflators of different countries. Hence, the constant 2005 PPP could be extended back and forth using the GDP deflator differential. Yet the weights on services and other sectors also change in the GDP deflator over time. Overall, time-varying PPPs have a number of appeals over constant PPPs. Nevertheless, they are plagued by various measurement problems. It is not clear which PPP would be best for MFP measurement.

increase the overall price level. At the same time, the Balassa-Samuelson effect is competitiveness neutral: PPP is assumed to hold for tradable goods. In other words, the real exchange rate based on tradable prices is assumed to be stable over time.

14.

A number of other channels can increase services prices (Baumol effect, Penn effect, Baghwati effect) and goods prices. For an overview, see e.g. Égert (2007).

15.

They usually come at three-or five-year intervals.

16. http://www.rug.nl/research/ggdc/data/pwt/v80/pwt_80_user_guide.pdf, Table 5.

Looking at different PPPs over time, a couple of striking features of the data emerge (Figure 4 below):

 There is for almost all countries a substantial wedge between constant 2005 PPP and other PPPs (chain-linked, GDP deflator and CPI-based PPPs)

 For some countries including Austria, Germany, France and Japan, the time-varying PPPs were above the 2005 PPP for most of the period under observation but the time-varying PPP measures were steadily converging to the 2005 PPP (Figure 4, panels A).

 For catching-up economies, the initially large negative gap is closing over time (Figure 4, panels B).

 For the last group of countries, there is no systematic gap between time-varying and constant 2005 PPP. The only exception is the CPI differential-extended PPP, which can deviate persistently from 2005 PPP (Figure 4, panels C). .28

.

32

.

36

.40

.

44

.

48

.

52

.56

.60

1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 est 400 500 600 700 800 900 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 kor 0 1 2 3 4 5 6 7 8 9 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 mex .

2

.

3

.4

.

5

.6

.7

.8

1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 1.6

1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 aus 1.16

1.20 1.24 1.28 1.32

1.36 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 can 0.85 0.90 0.95 1.00

1.05 1.10 1.15 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 fin .

52

.

56

.60

.64

.

68

.72

1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 chained PPP 2005 PPP extended using GDP deflator differential 2005 PPP extended using CPI differential constant 2005 PPP gbr 12.

A final question related to PPPs is at what rates can the capital stock be converted into the foreign benchmark's currency. The conversion cab be done using the rate applied to real GDP. Yet, the capital stock price level may be different from the overall price level. If productive capital is mainly composed of machinery and equipment and to the extent that these goods can be freely traded across countries, the conversion rate may be closer to the market exchange rate than the overall PPP. In such a case, applying the overall PPP rate to the capital stock may overstate the level of MFP (as it lowers the capital stock-to-GDP ratio). On the other hand, if the share of largely nontradable goods in the capital stock exceeds that in GDP, the use of the overall PPP rate will underestimate the true level of MFP. The difference between investment-specific PPP and overall PPP can be substantial. It is striking that investment PPPs in Central and Eastern Europe, Mexico and Turkey are systematically higher than the overall PPP. This implies that the relative price level of investment is closer to other countries', as expected, than the overall price level, largely influenced by the presence of non-tradable goods (Figure 5). Also, over time, the 2005 investment PPP extended by the investment deflator can show very large and persistent deviations from other PPP measures (Appendix B). Source: Author's calculations using data obtained from PWT8.0.

13.

In the MPF measures presented thus far, GDP was converted using 2005 PPPs (the ratio of the domestic and foreign absolute price levels in 2005) and the capital stock was converted using 2005 investment PPPs drawn from the Penn World Table 8.0. Two issues arise here. The first concerns whether investment-specific or economy-wide PPP conversion rates should be used for the capital stock. MFP25 shows that, as expected in line with Figure 5, applying economy-wide PPP to the capital stock reduces Turkey's MFP level for instance. But the difference is not very large. The second issue relates to the use of constant and time-varying PPPs. Using chained PPPs (MFP26) or 2005 PPPs adjusted for GDP deflator differences over time (MFP27) result in an unlikely pattern in Turkey's MFP series: very high at the beginning and constantly decreasing. This is the result of the hyperinflation Turkey experienced around 2000. This suggests that time-varying PPP measures should not be used in such circumstances (Figure 6).

Labour input

14.

Whether to use employment or total hours worked makes a smaller difference. The number of persons in employment (employees and self-employed) are used in the baseline OECD MFP calculations as a measure of labour input. This is something that can be measured with relatively high accuracy. Nevertheless, total employment measures may measure badly total labour input if the average yearly hours worked per employed person differs across countries (see e.g. OECD, 201). Such a divergence can happen for a number of reasons: i) regulation may put an upper ceiling on weekly hours worked such as France's 35-hours week work, ii) different number of bank holidays and the regulations regarding the level (and the 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 USA NOR CHE PRT ESP ITA JPN IRL TUR mfp27 evolution over a career) of holidays a person can take per year, iii) the share of part-time workers, and iv) the share of self-employed and entrepreneurs in total employment (as they can work as many hours as they wish). Using hours worked is desirable from a methodological and practical viewpoint. 17 However, changing employment to hours worked in MFP23 (including human capital) does not substantially change the overall profile of the series beside some shifts in relative levels (Figure 7). Total hours worked obtained from PWT8 (MFP232) are longer (e.g. Portugal) and available for more countries (e.g. Turkey) than those drawn from OECD countries (MFP23).

USA NOR CHE PRT ESP ITA JPN IRL TUR

Figure 7. Comparison of MFP measures with total employment (MFP21) and total hours worked (MFP23)

Note: MFP23: total hours worked obtained from the OECD Economic Outlook database, MFP232: total hours worked obtained from PWT8.0

Capital stock

15.

The way physical capital stocks are constructed can matter. There are three broad ways of measuring capital stock: i) using the perpetual inventory method (PIM), relying on investment and depreciation series, ii) using capital stock surveys, or iii) using balance sheet data. PIM is feasible in a wider set of countries, but it has limitations. First, the choice of the initial stock has to be based on some assumptions and it has a large influence on cross-country capital stock levels. Second, whether or not the depreciation rate is allowed to change over time can influence the level of, and the change in, the capital stock. The depreciation can change over time and in a cross-country comparison as well. Different types of capital may depreciate at different speed. If the composition changes over time and if it differs across countries, the depreciation rate will also change. In particular, investment in ICT has a much faster depreciation rate than other types of investment. Countries that started investing more heavily in ICT may have higher depreciation rates. More investment in ICT over time also implies increasing depreciation rates. These countries will also have different depreciation rates than countries that invest less in ICT.

17.

There are problems related to measuring hours worked by the self-employed are based on self-reporting.

Legally restricted weekly hours can be circumvented by supplementary contracts including a flat-rate remuneration of overtime work. Officially reported hours worked will not be able to capture this overtime work. But measurement problems might be even greater for total employment. Employment numbers for example will include those self-employed who work for example the bare minimum needed for inclusion, i.e. one hour a week. 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1

1 USA NOR CHE PRT ESP ITA JPN IRL TUR mfp232 16.
MFP12, MFP21, contain capital stocks with time-varying depreciation rates for all countries (from World Penn Table 8.0). MFP24 is calculated on the basis of capital stock series available from the OECD Economic Outlook database. For the core OECD countries, capital stocks are constructed using statistical surveys. 18 For the remaining countries, capital stocks are calculated using the PIM. 19 Using these data for level comparisons may be difficult: the base year used for individual countries can differ. A case in point is Turkey with a base year of 1998. Figure 8 shows how much lower Turkey's MFP series (MFP241) is compared to the other selected OECD countries. Rebasing all series to 2005 seems a solution. However, only real GDP can be rebased, as no capital stock deflators are available for the same series. The result, MFP 242 is depicted in Figure 8. There is little change for most countries (as they mostly have a base year of 2005). Nevertheless, the consequences are substantial for Turkey: rebasing GDP but not the capital stock, given the large gap between 2005 and 1998 raises MFP to a large extent. A more recent vintage of capital stock and real GDP series are based on 2010. The problem with 2010 is that the base year for all PWT series is 2005. Nevertheless, capital stock deflators are available for all but about five countries with different base years. Turkey is one of those, hence the additional increase in its MFP level (MFP 243). The picture remains broadly similar if 2010 PPP is used instead of the 2005 PPP to convert the series into a common currency. While the dynamics of the series may not change too much, it is clear that these series should not be used for cross-country comparisons.

17.

An easier way to handle different base years is to obtain country-specific MFP series domestic currency measures and the same base year for all components (output and capital stock). The MFP measure obtained can then be converted into a common currency using constant price PPPs. In this way, only the output (MFP) is converted, not the individual inputs.

18.

These countries are as follows: Australia, Austria, Belgium, Canada, Switzerland, Germany, Denmark, Spain, Finland, France, United Kingdom, Ireland, Italy, Japan, Korea, the Netherlands, New Zealand, Portugal, Sweden and the United States.

19. These countries are the following: Chile, Czech Republic, Estonia, Greece, Hungary, Island, Luxembourg, Norway, Poland, Slovakia, Slovenia, Turkey, Mexico and Israel. We rely on a set of plausibility checks, which would help to choose from the alternative MFP measures. MFP measures will be considered as realistic if they pass the following filters:

1. The top performers should include countries that plausibly belong to this group. In particular, the United States should be close to the frontier.

2. MFP in the countries close to the frontier should be broadly rising (and at least not declining) over decades. More generally, countries should not have a trend decrease in MFP over the sample period covering three decades.

19.

Based on these criteria, the measures MFP22 and MFP23 can be viewed as reasonable measures of MFP. MFP 12 to 17 suffer from the human capital bias: counterintuitive relative levels and trend decline for a number of countries. MFP24 faces the problem of cross-country comparability due to the different base years of the series, a major problem being Turkey. MFP25 appears OK in terms of criteria 1 to 3. Nevertheless, MFP 22 and MFP23 are more convincing conceptually: the capital stock should be converted 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1

USA NOR CHE PRT ESP ITA JPN IRL T UR m fp244
using capital stock-specific prices rather than economy-wide prices. Finally, MFP26 and MFP27, based on time-varying PPPs, show weird patterns for catching-up countries.

Concluding remarks

20.

Measuring multi-factor productivity correctly at the macroeconomic level and understanding its main drivers is key to assessing the impact of pro-growth reforms. Progress has been made in measuring multi-factor productivity (MFP) and understanding its determinants at the firm and industry levels (O' Mahony and Timmer, 2009; Syverson, 2011; Andrews and Cingalo, 2014; Bartelsman et al., 2014). Considerable work has been done in recent years to measure MFP but MFP is to some extent a measure of what we do not know. For instance, measures of MFP capturing the contribution of human capital within MFP may distort spatial and temporal comparisons. Indeed, the uncertainty about the precise level, comparability across countries, and the evolution over time has important implications for how precisely one is able to pin down the drivers of MFP in an empirical analysis. Not measuring accurately the MFP frontier masks our ability to reveal how far countries or firms are from the frontier and may lead to flawed conclusions regarding the speed of convergence to it. A wide margin of error may produce dubious results regarding the role of various institutions and policies for the evolution of MFP

21.

This paper carried out a systematic sensitivity analysis of MFP measurement by comparing a set of alternative MFP measures, with human capital included or excluded, with different measures of Purchasing Power Parity (PPP) exchange rates, using time-varying capital depreciation rates and different measures of capital stock and labour input (headcount against hours worked). The main result of the paper is that whether or not human capital is included in MFP makes a significant difference for the level and dynamics of MFP. At the same time, MFP measures are less sensitive to other parameters of the calculation. 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 .70

APPENDIX A

.75

.80

1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 .5

.6

.7

1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 1.12 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 chained PPP 2005 PPP extended using GDP deflator differential 2005 PPP extended using CPI differential constant 2005 PPP 2005 investment PPP extended using the investment deflator differential fra .

52

.

56

.60

.64

.68

.72 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 .6

.7

.8 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 .6

.7

.8

.9

1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 .6

.7

.8

1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1 chained PPP 2005 PPP extended using GDP deflator differential 2005 PPP extended using CPI differential constant 2005 PPP 2005 investment PPP extended using the investment deflator differential prt .

2

.

3

.4

.5

.6

.7

.8

1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 .6

.7

1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 1 9 8 5 1 9 8 7 1 9 8 9 1 9 9 1 1 9 9 3 1 9 9 5 1 9 9 7 1 9 9 9 2 0 0 1 2 0 0 3 2 0 0 5 2 0 0 7 2 0 0 9 2 0 1 1

  Figure 3. MFP measures excluding (MFP12) and including (MFP22) human capital

  Balassa-Samuelson effect is at work if productivity gains in the open sector exceeding those in the closed sector are higher in the domestic country compared to the foreign reference country. Wages in the open sector increase in line with productivity gain. If wages equalise across the open and closed sectors, wages will also rise in the closed sector. Wage increases in the closed sector, in turn, will translate into higher prices in the closed sector. Overall, productivity gains in the opens sector will raise services prices, which will

Figure 4 .

 4 Figure 4. Comparison of different PPPs over time, selected examples

Figure 5 .

 5 Figure 5. Comparison of overall and investment-specific PPPs in 2005

Figure 6 .

 6 Figure 6. Comparison of MFP measures (including human capital) and different PPP conversion rates

Figure 8 .

 8 Figure 8. Comparison of MFP measures with alternative capital stock measures

Figure

  Figure A1. Alternative MFP measures including human capital, 1985-2013

Figure B1 .

 B1 Figure B1. Comparison of the evolution and level of different PPP measures

  extended using the investment deflator differential fin Note: PPP_X Denotes chain-linked PPP, PPP_PGDP_X Is constant 2005 PPP extended using GDP deflator differentials, PPP_CPI_X Is the CPI-differential adjusted constant 2005 PPP and PPP_2005_X denotes constant 2005 PPP. IPPP_PIT_X is constant 2005 investment-specific PPP adjusted for the investment deflator differential.

  using GDP deflator differential 2005 PPP extended using CPI differential constant 2005 PPP 2005 investment PPP extended using the investment deflator differential hun using GDP deflator differential 2005 PPP extended using CPI differential constant 2005 PPP 2005 investment PPP extended using the investment deflator differential using GDP deflator differential 2005 PPP extended using CPI differential constant 2005 PPP 2005 investment PPP extended using the investment deflator differential

  Denotes chain-linked PPP, PPP_PGDP_X Is constant 2005 PPP extended using GDP deflator differentials, PPP_CPI_X Is the CPI-differential adjusted constant 2005 PPP and PPP_2005_X denotes constant 2005 PPP. IPPP_PIT_X is constant 2005 investment-specific PPP adjusted for the investment deflator differential.

  using GDP deflator differential 2005 PPP extended using CPI differential constant 2005 PPP 2005 investment PPP extended using the investment deflator differential pol

  using GDP deflator differential 2005 PPP extended using CPI differential constant 2005 PPP 2005 investment PPP extended using the investment deflator differential

Table 1 . Overview of alternative MFP definitions

 1

	1