International Mobility of Capital in the United States: Robust Evidence from Time-Series Tests

3.2.1 Standard OLSEG and RLS Estimates

The two-step OLSEG estimator sequentially involves the estimation of a static regression model in levels, Y(t)=α+βX(t)+ε(t), and then the estimation of an auxiliary, ε(t)=γε(t−1)+ν(t), or augmented auxiliary, Δε(t)=γε(t−1)+∑i=1kζ(i)Δε(t−i)+ν(t), regression to perform unit root tests on ε(t) and test H0: ε(t)∼I(1) (no cointegration among I(1) variables) against H1:ε(t)∼I(0) (cointegration among I(1) variables). The study estimates model [1] in levels and performs the ADF and PP unit root tests on ε(t) to test the null hypothesis of no cointegration against the alternative hypothesis of cointegration between saving and investment. The ADF test is performed using an autoregressive (AR) lag of k=2 (as suggested by the Akaike information criterion). The PP test is performed using the lag window (lw) of lw=4. The OLS estimates of model [1] and the ADF and PP test statistics for the null hypothesis of a unit root in ε(t) and the implied null hypothesis of no cointegration between saving and investment are as follows.

The figures in round parentheses are the t-ratios and in square brackets the 5 % critical values for the null hypothesis of a unit root in ε(t) for the ADF, PP (Davidson and Mackinnon 1993) and CRDW (Sargan and Bhargava 1983) tests. Both ADF and PP tests consistently reject H0: ε(t)∼(1) in favour of H1:ε(t)∼(0) and suggest the equilibrium relationship between saving and investment. ^[3] The use of 5 % critical values from Phillips and Ouliaris (1990) provides similar evidence. The magnitude of the long-run slope parameter on saving differs from both zero and unity in that the t-ratios reject both H0: β=0 and H0: β=1. The OLSEG estimates, thus, provide consistent support for the equilibrium relationship between the model series.

The equilibrium residuals measure the distance between the actual and forecast series of the model. Another way to test cointegration is to map the movements in the residual process of a cointegrating regression (Xiao 1999; Xiao and Phillips 2002). If the given series are cointegrated, then the residuals of a cointegrating regression should be stable with long-run movements within some critical bounds. Xiao (1999) and Xiao and Phillips (2002) argue that the cumulative sum (CUSUM) of recursive residuals test of Brown, Durbin, and Evans (1975) can be applied to the residuals of a regression to directly test the null hypothesis of cointegration. The study estimates the model using recursive least squares (RLS) to map the time-profile of the disequilibrium residuals and trace the temporal-trajectories of the model parameters. The time-trajectories of both CUSUM and CUSUM of squared residuals (CUSUMSQ) remain well within the 5 % critical bounds (Figure 1). The sequential F statistics are scaled by 5 % critical value such that the value larger than unity implies rejection and that less than unity the acceptance of the null hypothesis of model stability. The F statistics remain well below the critical unity grid and suggest the temporal stability of the recursive residuals, except for two spikes that cross the critical unity grid. The estimated residuals normally remain within the critical boundaries (Figure 1). The intercept, α, and slope, β, parameters move within the standard error confidence bands and seem temporally stable.

Figure 1:

Recursive least squares estimates and the temporal stability of the model.

3.2.2 Optimal Single-Equation Estimates

The standard OLS estimates become biased and inefficient in the presence of non-orthogonality of regressors and serial-correlation of residuals. The “super-consistency” property of OLS indeed allows one to omit I(0) regressors from the cointegrating model and asymptotically ignore the problems of endogeneity and serial-correlation. ^[4] In small samples, however, the OLS estimates remain biased and have inferential problems for the significance of long-run parameters. The bias is often substantial (Banerjee, Dolado, Galbraith, and Hendry 1993; Inder 1993) and the t statistics of cointegrating coefficients are generally not valid for statistical inference. The commonly used method to alleviate endogeneity is the IV or GMM estimator of Hansen (1982). The GMM has desirable properties in large samples. The standard moment conditions in a linear regression, y=Xβ+μ, require that E[μμ′]=Iσ2, E[X′μ]=E[X′(y−Xβ)]=0 and V[X′μ]=[X′X]σ2; where E is the expectation operator, X the matrix of exogenous variables and μ the vector of residuals. The OLS estimators are obtained by minimizing μ′μ=(y−Xβ)′(y−Xβ). The OLS parameter estimates should satisfy the moment condition in terms of the orthogonality (zero correlation) between exogenous variables and residual term, such that E[X′μ]=0. This moment condition, E[X′μ]=0, could either come from the assumptions about variables or/and be derived from the first-order conditions of an optimization problem. The condition is operationalized by replacing the expectation operator, E, by the sample average, T−1∑t=1TX′t(yt−Xtβ)=0.

If the orthogonality condition is not satisfied such that E[X′μ]≠0, then it is essential to replace X by some instrument set, Z, and minimize E[Z′μ] so that E[Z′μ]=0 and V[Z′μ]=[Z′Z]σ2; where matrix Z is assumed to have the same dimension as matrix X. The IV estimators, βˆIV=(Z′X)−1Z′y, are obtained by setting Z′μ=0 or Z′(y−Xβ)=0. If there are more moment conditions than the number of parameters, then the system of equations becomes algebraically over-identified and cannot be solved. This situation arises because the lagged (twice-lagged, thrice-lagged, …) variables tend to be the weak instruments, and it often becomes necessary to have a large set of moment conditions. Sargan (1958) develops the generalised instrumental variables estimator and suggests minimizing (y−Xβ)′Z(Z′Z)−1Z′(y−Xβ)=μ′Z(Z′Z)−1Z′μ. The matrix (Z′Z)−1 serves to weight the orthogonality conditions. The weighting matrix comes into operation when the dimensions of Z are larger than the dimensions of X. Hansen (1982) shows that a more efficient estimator can be obtained by replacing (Z′Z)−1 by the optimal weighting in terms of the inverse of the matrix, MCOV(zμ)=∑k=−LL∑t=1TZ′tμtμt−kZt−k. Hansen (1982) suggests the use of J∼χ2(n−p) statistic to test the moment conditions and examine the non-orthogonally of regressors to the residual process; where n is the number of instruments and p the number of parameters. The study performs the GMM estimation using the optimal weights suggested by Hansen (1982).

An optimal alternative to using GMM is to introduce an explicit AR(1) specification for X(t) along with the stochastic model for the relationship between Y(t) and X(t). The triangular representation of the cointegrated system of Phillips (1991), with I(1) series of Y(t) and X(t) and I(0) series of μ(t), suggests that the OLS estimator of β is consistent, but not generally fully efficient,

The asymptotic distribution of OLS estimator depends on various nuisance parameters engendered by serial-correlation in μ(t) and by correlation between μ(t) and innovation term for ΔX(t) in model [4]. The μ(t) and η(t) are cross-correlated not only contemporaneously, but also at various leads and lags. Phillips (1991) suggests using the following representation for μ(t),

The ξ(t) in model [5] is not correlated with η(t−j), ∀j∈[−k,k]. The cointegrating model [3] can be augmented with the leads and lags of ΔX(t) to resolve the problem of cross-correlation between μ(t) and η(t) (Phillips and Loretan 1991; Saikkonen 1991; Stock and Watson 1993). By substituting model [4] into model [5] and then substituting the resulting transform into model [3], the leads and lags cointegration estimator can be represented as

Since ξ(t) is not correlated with η(t) in model [5], it will also be uncorrelated with ΔX(t) in model [6]. The ΔX(t) asymptotically eliminates the effect of endogeneity of X(t) on the distribution of OLS estimator of β. If ξ(t) is independently and identically distributed, then the standard distribution theory can be used to perform inference on OLS parameter estimates. While the leads and lags of ΔX(t) resolve the problem of endogeneity of X(t), they do not necessarily eliminate all the serial-correlation and heteroscedasticity in ξ(t). Stock and Watson (1989, 1993) suggest using GLS to estimate model [6]. The GLS estimates of standard errors and variance-covariance matrix could be used to construct the asymptotically valid χ² hypothesis tests on β. Phillips and Loretan (1991) argue that due to persistence in the effects of innovations arising from unit roots in the system, the lags of ΔX(t) are generally not an adequate proxy for the past history of μ(t) and they suggest using a parametric correction in model [6] to account for the potential serial-correlation in ξ(t). The requisite information set for valid conditioning is better modelled by using lagged equilibria than by using lagged differences of the dependent variable, and they recommend augmenting model [6] with the lagged levels of [Y(t)−α−βX(t)],

The ζ(t) is serially uncorrelated and model [7] can be estimated using non-linear least squares (NLLS). The NLLS estimator of β is asymptotically efficient and the estimates of variance-covariance matrix have the standard limiting distribution. The NLLS variance-covariance matrix can be used to perform the hypothesis test on β in a standard manner.

Phillips and Hansen (1990) develop the fully-modified OLS (FMOLS) estimator to resolve the problems of endogeneity-bias and serial-correlation, and obtain the efficient estimates of the model parameters. The FMOLS estimator starts with the standard OLS regression and then, analogous to the Phillips-Perron (Phillips and Perron 1988) unit root test, makes a non-parametric correction to account for the endogeneity-bias and serial-correlation that may show up in the OLS residuals. The estimates of the long-run parameters and the associated t-statistics are, thus, adjusted to correct for the bias arising from the endogeneity of regressors and serial-correlation of residuals. The FMOLS estimator is super-consistent and is asymptotically both unbiased and normally distributed (Park and Phillips 1988; Phillips and Hansen 1990; Hansen and Phillips 1990). The t-statistics of the long-run coefficients are asymptotically normally distributed, and the standard limiting distributions can be used to perform the statistical inference in the FMOLS estimates.

The DOLS and NLLS estimations require the determination of the optimal lags and leads structures of the dynamic regressors. The Akaike information criterion (AIC) pointed towards an unduly large lags and leads structure of k{–9, 0, +9}, while the Schwarz information criterion (SIC) suggested a relatively parsimonious structure of k{–4, 0, +4} for the DOLS estimator. The model with DOLS is estimated using the parsimonious lags and leads structure of k{–4, 0, +4} as suggested by SIC. The AIC and SIC consistently suggested k{–1, 0, +1} for the model with NLLS, and the Ljung-Box portmanteau (LB-Q) statistic (Ljung and Box 1978) did not reject the null hypothesis of no serial-correlation in the model residuals. The NLLS estimation is, therefore, carried out using k{–1, 0, +1}. The estimations are also carried out using the higher lags and leads structure of k{–5, 0, +5} for the DOLS and k{–2, 0, +2} for the NLLS estimator so as to assess the robustness of results. The standard errors of the parameters from DOLS and NLLS estimators are adjusted using the heteroscedasticity and autocorrelation consistent (HAC) estimator of Newey and West (1987). The GMM, DOLS, FMOLS and NLLS estimates of the model consistently suggest the positive and significant long-run effects of domestic saving on investment (Table 3). The J-statistics in GMM do not reject the null hypothesis of no correlation between the regressors and the residual term. The long-run slope parameter on saving ranges between 0.31 and 0.37, and the t-ratios reject both H0: β=0 and H0 : β=1 at 1 % level across all the estimators. The numerically low magnitudes of the slope parameter on saving (0.31–0.37), consistently across all the estimators, point towards the high mobility of capital. The results are robust to the variations in instrument set in GMM and the model structures in DOLS, FMOLS and NLLS estimations (see Annexure 1).

The conventional CUSUM and CUSUMSQ tests used to map the time-profile of the disequilibrium residuals become biased and inefficient when the model residuals are autocorrelated and regressors are characterised by endogeneity. Xiao and Phillips (2002) use the FMOLS estimator to resolve the problem of serial-correlation and endogeneity. They construct the cumulative sum (CS_n) and moving sum (MS_n) test statistics to test the direct null hypothesis of “cointegration” against the alternative hypothesis of “no cointegration” among I(1) variables. The study uses the optimal residuals obtained from all the GMM, DOLS, FMOLS and NLLS estimates of the long-run model to construct the cumulative sum, CSn=maxk=1,...n1/ω^u,x2 n|∑t=kε^t|, and moving sum, MSn=maxk=1,...n−[nh]1/1ωˆu,x2nωˆu,x2n∑t=kk+[nh]εˆt, test statistics and perform the new cumulative sum (CUSUM) and moving sum (MOSUM) tests; where εˆt represents the optimal residuals of the equilibrium relationship among the model variables, ωˆu,x2 is a semi-parametric kernel estimator of ωu,x2 and 0<h<1 is the bandwidth parameter for the moving window (see Xiao 1999; Xiao and Phillips 2002). The new CUSUM and MOSUM tests, based on optimal residuals, overcome the problem of endogeneity of regressors and serial-correlation of residuals and provide efficient estimates, as compared to the conventional CUSUM and CUSUMSQ tests based on the standard OLS recursive residuals. The results suggest that the CSn statistics do not reject the direct null hypothesis of cointegration and cross-validate the equilibrium relationship between saving and investment (Table 3). ^[5]

3.2.3 Maximum-Likelihood System Estimates

The maximum-likelihood (ML) system estimator of Johansen (1991) estimates the k^th order vector autoregression model and takes a system-based account of endogeneity.

Model [8] can be reparameterized as

The Γ(i)=−I−∑i=1k−1Π(i),Π˜=−I−∑i=1kΠ(i) and Π=αβ′ in model [9]. The X′=[I/IY,Y,S/SY]Y] is a p × 1 vector of p number of I(1) variables, μ is a vector of constants, and ε(t) is a p-dimensional vector of disturbances with zero-mean and covariance matrix Σ [i. e. ε(t) ~ (0, Σ)]. Model [9] is estimated using k=2, as suggested by the likelihood ratio (LR) test (Sims 1980).

The asymptotic λ-trace and the λ-trace adjusted for small-sample (Johansen 2000, 2002) consistently reject the null hypothesis of r=0 (but not r ≤ 1) and suggest the presence of one cointegrating vector (Table 4). The long-run parameter on S/Y is statistically significant and dimensionally consistent with the estimates obtained from the single-equation estimators. The χ2 statistic rejects the null hypothesis of zero-restriction on the parameter of S/Y and suggests the significant long-run effects of domestic saving on investment. The results are, however, sensitive to the use of lag structures in the VAR model. Both asymptotic λ-trace and λ-trace adjusted for small-sample reject the null hypothesis of no cointegration in the models estimated with lags k=1 and k=2, but not in the models estimated with lags k=3, k=4 and k=5 (Table 4). The presence of cointegration does not necessarily imply that the estimated parameters are temporally stable. The test of parameter constancy, based on recursive estimation, is performed to map the temporal stability of the cointegrating vector (Hansen and Johansen 1993, 1999). The recursive estimation is carried out starting with a base sample of 1950 and then sequentially increasing the sample through 2007. The recursive test statistics, X(t) and R1(t), are scaled by 5 % critical value, such that the value less than unity implies the acceptance and that larger than unity the rejection of the null hypothesis hypothesis of constancy of β vector. Both X(t) and R1(t) statistics remain well below the critical unity grid, and do not reject the null hypothesis of constancy of β vector (Figure 2).

Figure 2:

ML estimates and the recursive test of beta constancy.

3.3.1 Vector Error-Correction Estimates

The estimates of the cointegrating model show only the steady-state equilibrium relationship and do not provide any information on the short-run dynamics. The Granger Representation Theorem (Engle and Granger 1987) postulates that if y(t) and x(t) sequences of a given regression in levels, y(t)=α+βx(t)+ε(t), are I(1) and are cointegrated, then Δy(t)=y(t)−y(t−1), Δx(t)=x(t)−x(t−1) and the common stochastic process ε(t)=y(t)−α−βx(t) are all I(0) and, in such case, there exists a valid error-correction representation of the time-series. The error-correction model resolves the problem of “spurious regression” without losing long-run information contained in the level variables. The study estimates the vector error-correction model (VECM) to measure the speed-of-adjustment towards steady-state equilibrium and test the null hypothesis of Granger non-causality.

The z(t–1) is the lagged error-correction term and αi∈[0,−1],∀i=1,2, measures the speed-of-adjustment towards steady-state equilibrium. The lagged error-correction term, z(t–1), in model [10] and model [11] is obtained from the cointegrating vector of the long-run model estimated using alternatively the OLSEG, DOLS, FMOLS, NLLS and ML estimators. The higher (lower) the speed-of-adjustment, the lower (higher) would be the dispersion from long-run equilibrium and implied lower (higher) would be the mobility of capital. The significant parameter of the lagged error-correction term is relatively a more efficient method for rejecting the null hypothesis of no cointegration and determining the equilibrium relationship, as compared to the static OLSEG which does not account for the model dynamics (Kremers, Ericsson, and Dolado 1992). The standard limiting distributions can be used to test the significance of the error-correction parameter. The first-differenced dynamic regressors and z(t−1) are the two possible sources of non-causality in VECM. The saving causes investment, if H0:∑i=1kλ(i)=0 or H0:α1=0 in model [10] is rejected, but H0:∑i=1kς(i)=0 or H0:α2=0 in model [11] is not rejected. The investment causes saving, if H0:∑i=1kς(i)=0 or H0:α2=0 in model [11] is rejected, but H0:∑i=1kλ(i)=0 or H0:α1=0 in model [10] is not rejected. The saving and investment are characterised by bi-directional causality, if H0:∑i=1kλ(i)=0 or H0:α1=0 in model [10] and H0:∑i=1kς(i)=0 or H0:α2=0 in model [11] are rejected. The short-run dynamic models [10] and [11] in first-differences are augmented with the lagged error-correction term, z(t–1), obtained from the long-run model estimated alternatively with (i) OLSEG (Model I), (ii) DOLS (Model II), (iii) FMOLS (Model III), (iv) NLLS (Model IV) and (v) ML (Model V) estimators, and, as such, five variants of VECM are estimated to test the null hypothesis of Granger non-causality and take an encompassing account of the adjustment process towards steady-state equilibrium (Table 5).

The results remain consistent across all the VECMs, and suggest the significant effects of lagged disequilibrium (Model I to Model V, Table 5). The VECMs reinforce the evidence obtained from the long-run models and point towards the equilibrium relationship between domestic saving and investment. The t-statistics reject H0:α1=0 consistently across all the estimates, and cross-validate the cointegrating relationship. The speed-of-adjustment towards steady-state equilibrium is quite fast, ranging between 60 % to 63 % per annum. The implied departure from equilibrium {1−α} ranges between 37 % to 40 %, and it points towards the high mobility of capital. The Jarque−Bera∼χ2 statistic does not reject the null hypothesis of normality in favour of the non-normal alternative. The VECMs consistently suggest bi-directional causality with stronger evidence for the effects of saving on investment, as compared to the feedback effects of investment on saving. The causality comes mainly from the “disequilibrium adjustments,” rather than the “short-run dynamics.”

The impulse response and variance decomposition analyses are carried out on all the VECMs (Model I to Model V) to trace the time-profile and map the response trajectories of ΔI/IYYΔS/SYY to the shocks to the innovations of ΔS/SYYΔI/IYY. The results for the impulse response functions and variance decomposition analyses remain consistent across all the VECMs and are, therefore, reported only for one VECM based on the cointegrating vector obtained from the ML system estimator (Model V, Table 6). The impulse response functions are constructed using the estimated parameters and, since each parameter is estimated imprecisely, these functions also contain errors (Enders 2004). The standard error bands around the impulse response functions are constructed to account for the parameter uncertainty inherent in the estimation process and map the width of the prediction intervals. The Monte Carlo simulations are performed using 10,000 draws and the analysis is carried out for a time horizon of 10 years. Both impulse response functions and decompositions of variance consistently suggest that ΔI/IYY [DGDIRATE] and ΔS/SYY [DGDSRATE] show larger responses to the shocks to their own innovations than to the shocks to the cross innovations of other series in the system (Figure 3, Table 6). The innovations of DGDSRATE explain around 12 % of the variations in the innovations of DGDIRATE, while the innovations of DGDIRATE explain around 40 % of the variations in the innovations of DGDSRATE. The innovation accounting reinforces the evidence obtained from VECM and suggests the bi-directional causality between saving and investment. The effects of saving on investment seem stronger than the feedback effects of investment on saving.

Figure 3:

Impulse response functions and the standard error bands (based on ML-based VECM; VAR lag k=2).

3.3.2 Over-Parameterized Level-VAR Estimates

The VECM approach to testing Granger non-causality builds sequentially on the (i) pretesting of model series for unit root(s), (ii) determining of cointegration rank and (iii) eventually the testing of zero-restrictions on the parameters of the short-run dynamic and long-run lagged disequilibrium regressors. The unit root tests used to determine the I(d) properties of the model series are known to have low power in small samples. The subsequent cointegration tests conditioned on the unit root pre-tests and that non-causality tests conditioned sequentially on the cointegration pre-tests could potentially contain a pre-test bias. The over-parameterized level-VAR estimator of Toda and Yamamoto (1995) (TY) by-passes all the pretesting requirements and is applicable to all VAR systems characterised by a stationary (around a deterministic trend) or integrated or cointegrated process of an arbitrary order. The TY estimator involves the (i) estimation of the level-VAR model of order k augmented artificially with extra d_max lags, and then the (ii) testing of zero-restrictions on the parameters of first k lagged (but not all lagged) regressors to draw long-run Granger causal inference; where d_max is the maximal order of integration of the system series. The VAR lag-length, k, can be determined using the usual model selection criteria and lag-length selection tests.

The ω is a vector of intercept terms, ∑i=1q=k+dmaxξ(i) a matrix of long-run parameters and ε a vector of white-noise residuals with usual Gaussian iid properties. The d_max lag(s) augmentation represents the artificial over-loading of the true lag-length, k, and the implied over-parameterization of the level-VAR model. The level-VAR model [12] can be reparameterized as

Under the null hypothesis of zero-restrictions on the parameters of each variable in vector X′=[I/IY,Y S/SY]Y] for lags one to k, the Wald statistic has an asymptotic χ2 distribution with usual degrees of freedom. The AIC suggested lags k=5, while the SIC suggested lags k=2 for the VAR model. The study uses lags k=2 (as suggested by SIC) and estimates three sets of over-parameterized level-VAR models; (i) one based on dmax=0 and implied q=[k + dmax]=[2 + 0]=2 lags, (i) second based on dmax=1 and implied q=[k + dmax]=[2 + 1]=3 lags, and (iii) third based on dmax=2 and implied q=[k + dmax]=[2 + 2]=4 lags of the VAR model. The estimates of these over-fitted VAR models are used to compute the joint F-statistics for the null hypothesis of zero-restrictions on the parameters of regressors up to first k=2 lags (ignoring extra dmax lags loadings). The over-parameterized level-VAR models are also estimated using k=3 lags and implied (i) q=[k + dmax]=[3 + 0]=3 lags, (ii) q=[k + dmax]=[3 + 1]=4 lags and (iii) q=[k + dmax]=[3 + 2]=5 lags of the VAR model so as to determine the robustness of results.

The joint F statistics computed for first k=2 lags [in the model with q=[k + d_max]=[2 + 0]=2 lags] reject the null hypothesis of zero restrictions on the parameters of the lagged regressors of (i) S/SYYt in the model with I/IYYt as the regressand and (ii) I/IYYt in the model with S/SYYt as the regressand (Table 7). The F statistics for first k=3 lags [in the model with q=[k + d_max]=[3 + 0]=3 lags] reject the null hypothesis (at 10 % level) of zero restrictions on the parameters of the lagged regressors of S/SYYt in the model with I/IYYt as the regressand. The zero-restrictions are not rejected for the parameters of the lagged regressors of I/IYYt in the model with S/SYYt as the regressand. The F statistics do not reject the null hypothesis of zero restrictions on the parameters of the distributed lagged regressors of (i) S/SYYt in the model with I/IYYt as the regressand and (ii) I/IYYt in the model with S/SYYt as the regressand in all the remaining models with over-parameterizations based on dmax=1 and dmax=2 (see off-diagonal elements, Table 7). The zero restrictions are rejected (at 1 % level) for the parameters of the autoregressive regressors in all the models estimated with over-parameterizations set at dmax=0, dmax=1 and dmax=2 (see diagonal elements, Table 7). The sensitivity analysis undertaken further to ascertain the robustness of results suggests that the models estimated using the higher lag structures of k=4 and k=5 consistently do not provide any support for the long-run Granger-causal nexus between domestic saving and investment (see Annexure 2). It needs to be recognised that the TY estimator suffers from the loss of power and tends to be inefficient, given that it is based on an intentionally over-fitted VAR system.

3.4.1 Standard Tests for Model Instability

The parameter instability tests of Hansen (1992) test the null hypothesis of constant parameters against the alternative that the parameters follow a martingale. Hansen (1992) develops the test statistics denoted as (i) L to test the stability of the individual coefficients of a regression model, (ii) Lc to test the joint stability of the regression coefficients and (iii) σε2 to test the stability of the regression residuals. ^[6] These tests are valid for both static and dynamic regression models. The Lc statistic is asymptotically robust to heteroscedasticity. The OLS estimates of the model juxtaposed with the Hansen test statistics for the null hypothesis of parameter stability are as follows.

Rˉ2=0.54; Hansen Joint Lc=0.90 {0.08}; Hansen σε2=0.11{0.54}