Truncated Product Methods for Panel Unit Root Tests

Abstract

This paper proposes two new panel unit root tests based on Zaykin et al. (2002)’s truncated product method. The first one assumes constant correlation between p-values and the second one uses sieve bootstrap to allow for general forms of cross-section dependence in the panel units. Monte Carlo simulation shows that both tests have reasonably good size and are powerful in cases of some very large p-values. The proposed tests are applied to a panel of real GDP and inflation density forecasts, resulting in evidence that professional forecasters may not update their forecast precision in an optimal Bayesian way.

1 Introduction

Recently, advances in panel unit root studies that provide reliable inference despite cross-section dependence have spurred interest in testing for unit roots in macroeconomic data.¹ O’Connell (1998) considered a GLS-based unit root test for homogeneous panels. Chang (2004) showed that this GLS procedure depends on nuisance parameters and proposed a corrective bootstrap approach. Phillips and Sul (2003), Bai and Ng (2004), Moon and Perron (2004) and Pesaran (2007) proposed dynamic factor models that allow the common factors to have different effects on cross-section units. For a review on these so-called “second generation” panel unit root tests, see Breitung and Pesaran (2008) and Gengenbach et al. (2010).

We propose new methods for panel unit root tests by combining dependent p-values. The p-value combination methods were introduced to panel unit root literature independently by Maddala and Wu (1999) and Choi (2001). Recent contributions include Demetrescu et al. (2006) and Hanck (2008). Compared to combining test statistics, combining p-values has several advantages in that it allows different specifications for each panel unit, any unit root test derived and unbalanced panels.

Our proposed tests are based on Zaykin et al. (2002)’s truncated product method (TPM), which has been widely used in biostatistics, see Schmidt et al. (2008) and Seebacher and Glanville (2010). The TPM takes the product of the p-values below some pre-specified cut-off value, increasing power in cases of some very large p-values. Building on the TPM, we propose two panel unit root tests: a modified TPM that assumes a constant correlation among the p-values and a bootstrap TPM that allows for general forms of cross-section dependence in the panel units. Monte Carlo simulation provides evidence of good power with moderate and large T values for both tests, despite the slightly oversized modified TPM.

For an empirical example, we test the null hypothesis that forecast precision, if perceived properly, should contain a unit root, as implied by the Bayesian learning model developed in Lahiri and Sheng (2008). Based on a panel of density forecasts for real GDP and inflation during 1992–2009, we find evidence that some professional forecasters do not optimally update their forecast precision.

2 Truncated Product Method

Consider the model

$$y_{it}=(1-{\alpha }_i){\mu }_i+{\alpha }_i{y}_{i,t-1}+{\epsilon }_{it},i=1,\dots ,N;t=1,\dots ,T.$$ (1)

The specification in equation (1) allows for heterogeneity in both the intercept and the slope. For convenience, equation (1) is rewritten as

$$\mathrm{\Delta }{y}_{it}=-{\phi }_i{\mu }_i+{\phi }_i{y}_{i,t-1}+{\epsilon }_{it},$$ (2)

where Δy_it = y_it − y_i,t−1 and φ_i = α_i − 1. We test

$$H_0:{\phi }_i=0\text{for}\text{all}i$$ (3)

against the alternative

$$H_1:{\phi }_i<0,i=1,\dots ,N_0;{\phi }_i=0,i=N_0+1,\dots ,N,$$ (4)

such that

$$\underset{N\to \infty }{lim}\frac{N_0}{N}=\delta ,0<\delta \le 1.$$ (5)

Remark 1

Note that the null and alternative hypotheses can also be written as H₀: δ = 0 vs. H₁: δ > 0. Thus, rejection of the null can be interpreted as evidence for rejecting the unit root hypothesis for a non-zero fraction of panel units as N → ∞.

Let S_i,Ti be a test statistic applied to the ith unit of the panel in equation (2). Then the corresponding p-value is defined as p_i = F(S_i,Ti), where F(·) denotes the cumulative distribution function (c.d.f.) of S_i,Ti. We assume

Assumption 1

(Uniformity) Under H₀, S_i,Ti has a continuous distribution function, which ensures a uniform distribution of the p-values.

Zaykin et al. (2002) suggested the use of the product of all those p-values that do not exceed some pre-specified value τ such that

$$W=\prod _{i=1}^N{p}_i^{I(p_i\le \tau )},$$ (6)

where I(·) is the indicator function.

Remark 2

When some series in the panel are clearly nonstationary such that the resulting p-values are close to 1, traditional p-value combination methods may lose power because they could be dominated by these large p-values. By truncating, these large components are removed, thereby providing more power, much like how a “trimmed mean” gains efficiency in the presence of outliers.

When all p-values are independent, W has a closed form distribution, see Zaykin et al. (2002). However, the distribution of W is unknown when the independency assumption is violated. Motivated by the success of the TPM in biostatistics, below we extend the TPM to allow for cross-section dependence among the p-values in panel unit root tests.

2.1 Modified TPM

First we modify the TPM, denoted by W_m, by assuming a constant correlation between the p-values. Although constant correlations may not hold in some empirical applications, W_m at least does not require the panel to be balanced. The procedure is as follows:

• Step 1:

  Calculate W_m using equation (6).

• Step 2:
  Estimate the correlation matrix, Σ, for p-values. Following Hartung (1999) and Demetrescu et al. (2006), we assume a constant correlation between the probits t_i and t_j,

  $$cov(t_i,t_j)=\rho ,\text{for}i\ne j,i,j=1,\dots ,N,$$
  where t_i = Φ⁻¹(p_i) and t_j = Φ⁻¹(p_j). ρ can be estimated by

  $$\stackrel{\sim }{\rho }=max(-\frac{1}{N-1},\widehat{\rho }),$$

  where $\widehat{\rho }=1-\frac{1}{N-1}{\sum }_{i=1}^N{(t_i-\overline{t})}^2$ and $\overline{t}=\frac{1}{N}{\sum }_{i=1}^N{t}_i$.

• Step 3:

  The distribution of W_m is calculated based on the following Monte Carlo simulations:

  1. Draw pseudo-random probits from the normal distribution with mean zero and the estimated correlation matrix, Σ̂, and transform them back through the standard normal c.d.f., resulting in N p-values, denoted by p̃₁, p̃₂, ···, p̃_N.

  2. Calculate $\stackrel{\sim }{W}={\displaystyle \prod _{i=1}^N}{\stackrel{\sim }{p}}_i^{I({\stackrel{\sim }{p}}_i\le \tau )}$.

  3. Repeat steps 3a and 3b B times and get W̃_l, l = 1, …, B.

  4. The p-value for W_m is given by $p=\frac{1}{B}{\sum }_{l=1}^{B}I({\stackrel{\sim }{W}}_l\le W_m)$.

Remark 3

Note that the transformation marginally alters the correlation among p-values. However, as pointed out by Zaykin et al. (2002), the correlation is approximately invariant under monotone transformations. Therefore, the correlation between the probits t_i and t_j should be roughly equal to the correlation between the p-values p_i and p_j.

Remark 4

We circumvent the problem of a degenerate correlation matrix Σ in the case for T < N by applying Hartung (1999)’s proposal, thus allowing for cases of very large N.

2.2 Bootstrap TPM

To preserve the dependence structure among cross-section units, we extend Palm et al. (2008)’s bootstrap method by resampling entire cross sections of residuals. We make the following assumptions:

Assumption 2

(Linearity) The error term ε_it is given by a general linear process

$${\epsilon }_{it}={\psi }_i(L)e_{it},$$ (7)

where ${\psi }_i(z)={\sum }_{k=0}^{\infty }{\psi }_{ik}{z}^k$ and ${\sum }_{k=0}^{\infty }\mid {\psi }_{ik}\mid <\infty$ for i = 1, …, N.

Assumption 3

(Dependency; see also Chang (2004) Assumption 1) Define the N × 1 vector e_t ≡ (e_1t, …, e_Nt)′ for t = 1, …, T. Let e_t be a sequence of i.i.d. random variables such that Ee_t = 0, ${E\mathit{e}}_t{\mathit{e}}_t^{\prime }=\mathbf{\sum }$ and E||e_t||⁴ < ∞, where ||·|| is the Euclidean norm.

Below we outline the necessary steps for conducting the bootstrap TPM, W^*.

• Step 1:

  Calculate W^* using equation (6).

• Step 2:
  Obtain the residuals from an Augmented Dickey-Fuller (ADF) regression

  $${\widehat{e}}_{it}=y_{it}-{\widehat{\alpha }}_i{y}_{i,t-1}-\sum _{j=1}^{J_i}{\widehat{\phi }}_{ij}\mathrm{\Delta }{y}_{i,t-j},$$ (8)

  where the lag order J_i is selected according to the modified AIC procedure by Ng and Perron (2001). Then form the time series residual vectors ê_t ≡ (ê_1t, …, ê_Nt)′ for t = 1, …, T.²

• Step 3:

  Generate the N × 1 vector ${\mathit{e}}_t^{\ast }\equiv {\left(e_{1t}^{\ast },\dots ,e_{Nt}^{\ast }\right)}^{\prime }$ by resampling from the centered residual vectors $\left({\widehat{\mathbf{e}}}_t-T^{-1}{\sum }_{t=1}^T{\widehat{\mathbf{e}}}_t\right)$, t = 1, …, T. The bootstrap samples ${\mathit{e}}_t^{\ast }$ constructed as such will preserve the cross-section dependence structure of the data, as pointed out by Maddala and Wu (1999).

• Step 4:
  Generate ${\epsilon }_{it}^{\ast }$ recursively from $e_{it}^{\ast }$ as

  $${\epsilon }_{it}^{\ast }=\sum _{j=1}^{J_i}{\widehat{\phi }}_{ij}{\epsilon }_{i,t-j}^{\ast }+e_{it}^{\ast }.$$ (9)
• Step 5:
  Impose the null of unit root to obtain bootstrap samples $y_{it}^{\ast }$ as

  $$y_{it}^{\ast }=y_{i,t-1}^{\ast }+{\epsilon }_{it}^{\ast }.$$ (10)

  We set $y_{i,-50}^{\ast }=0$ and run the recursion for 50 initial observations before using $y_{it}^{\ast }$ to mitigate the effect of initial conditions.

• Step 6:

  Based on the bootstrap sample $y_{it}^{\ast }$, calculate W̃, defined in equation (6).

• Step 7:

  Repeat steps 3–6 B times and get W̃_l, l = 1, …, B.

• Step 8:

  The p-value for W^* is given by $p=\frac{1}{B}{\sum }_{l=1}^{B}I({\stackrel{\sim }{W}}_l\le W^{\ast })$.

3 Monte Carlo Study

Initially we consider dynamic panels with a common factor driving the cross-section dependence. The DGP is as in equation (1), where

$${\epsilon }_{it}={\gamma }_i{f}_t+{\xi }_{it},$$ (11)

for i = 1, …, N, t = −50, −49, …, T with the initial value y_i,−50 = 0. The factor loading γ_i is drawn from a uniform distribution as γ_i ~ i.i.d. U[0, 3]. The individual fixed effect μ_i, the common factor f_t and the error term ξ_it are independently drawn from normal distributions as μ_i ~ i.i.d. N(0, 1), f_t ~ i.i.d. $N(0,{\sigma }_f^2)$ and ξ_it ~ i.i.d. N(0, 1). We explore the properties of the tests under cross-section independence with ${\sigma }_f^2=0$ (DGP 1) and under “high” cross-section dependence with ${\sigma }_f^2=10$ (DGP 2).

Remark 5

Under the null hypothesis of common and idiosyncratic unit roots, the DGP setup in Bai and Ng (2004) is equivalent to our DGP in (1) and (11) when α_i = 1 for all i. However, these two DGPs are different in the case of a unit root in the common factor and near-unit roots in the idiosyncratic errors. See Banerjee et al. (2004) for a detailed description of this case of cross-unit cointegration.

Next we allow for serial correlation in the error terms. We consider a number of experiments where the errors ξ_it in (11) are generated either as an AR(1) process ξ_it = ρ_iξ_i,t−1 + e_it (DGP 3), or as an MA(1) process ξ_it = e_i,t + λ_ie_i,t−1 (DGP 4), where e_it ~ i.i.d. N(0, 1). We choose ρ_i ~ i.i.d. U[0.2, 0.4] or U[−0.4, −0.2] and λ_i ~ i.i.d. U[0.2, 0.4] or U[−0.4, −0.2]. These DGPs are intended to check the behavior of our tests under different types of serial correlation.

Finally, we consider spatial dependence as an alternative scenario of panel cross-section dependence. Following Baltagi et al. (2007), we consider two commonly used spatial error processes: the spatial autoregressive (SAR) and the spatial moving average (SMA). The SAR specification (DGP 5) for the N × 1 error vector ε_t in (1) can be expressed as ε_t = (I_N − θ₁W_N)⁻¹υ_t, where W_N is a known N × N spatial weights matrix. θ₁ is the spatial autoregressive parameter and the error component υ_t is assumed to be distributed independently across a cross-section dimension with constant variance ${\sigma }_v^2$. In contrast, the SMA specification (DGP 6) for the error vector ε_t can be expressed as ε_t = (I_N + θ₂W_N)υ_t, where θ₂ is the spatial moving average parameter. Without loss of generality, we let ${\sigma }_v^2=1$. We consider the spatial dependence with θ₁ = 0.8 and θ₂ = 0.8. We specify the spatial weight matrix W_N as a “1 ahead and 1 behind” matrix with the ith row (1 < i < N) of this matrix comprising of nonzero elements in positions i + 1 and i − 1. Each row of this matrix is normalized such that all non-zero elements are equal to 1/2.

For all of DGPs considered here, we choose

$${\alpha }_i\{\begin{array}{ll}~\text{i}.\text{i}.\text{d}.U[0.85,0.95]\hfill & \text{for}i=1,\dots ,N_0,\text{where}{N}_0=\delta ·N\hfill \\ =1\hfill & \text{for}i=N_0+1,\dots ,N.\hfill \end{array}$$

The value of δ indicates the fraction of stationary series in the panel, varying in the interval 0–1. When δ = 0, we explore the size of the tests. Choosing δ = 0.1, 0.5 and 0.9, we analyze the impact of the proportion of stationary series on the power of the tests. The tests are one-sided with the nominal size set at 5%, and conducted for all combinations of N ∈ {20, 50} and T ∈ {20, 50, 100}. The results are obtained with MATLAB using M = 2000 simulations. Within each simulation, additional B = 1000 bootstrap replications are performed.

We calculate the ADF t statistics. The number of lags in the ADF regressions is selected according to the modified AIC procedure suggested by Ng and Perron (2001). We obtain p-values of unit root tests using response surface regressions as provided by MacKinnon (1996). ³

As a preliminary check, we compute the pairwise cross-section correlation coefficient, ρ̂_ij, of the residuals from the ADF regressions. Following Pesaran (2004), we construct the average of these correlation coefficients as $\overline{\widehat{\rho }}=\frac{2}{N(N-1)}{\sum }_{i=1}^{N-1}{\sum }_{j=i+1}^N{\widehat{\rho }}_{ij}$ and the associated cross-section dependence (CD) test statistics as $CD=\sqrt{\frac{2T}{N(N-1)}}{\sum }_{i=1}^{N-1}{\sum }_{j=i+1}^N{\widehat{\rho }}_{ij}$, where CD ~ N(0, 1) under the null of no cross-section dependence. When the null hypothesis is rejected, we use our proposed tests; if not, we use Zaykin et al. (2002)’s original W test. The average correlation coefficient is 0 under cross-section independence (DGP 1), between 3% to 22% under spatial dependence (DGP 5 and 6) and about 80% with a factor structure (DGP 2–4). Thus our considered DGPs represent a wide range of cross-section dependence. The CD statistics reject the null and prompt us to use W_m and W^* in all cases except DGP 1.

Below we report the size and power of the modified TPM (denoted by W_m) and bootstrap TPM (denoted by W^*). For comparison, we also include some other commonly used panel unit root tests. ⁴ In the absence of clear guidance for the choice of τ, we try 10 different cut-off values, ranging from 0.05, 0.1, 0.2, ···, up to 0.9. We find that both original and proposed TPMs tend to have relatively small size distortions with a smaller τ, and that their power does not show any clear patterns. We only present the results for τ = 0.1 because our simulation results are similar for 0.05 ≤ τ ≤ 0.2. ⁵

With no cross-section dependence (not reported here), the P, S, W and CIPS tests yield good empirical size. The other tests are slightly undersized. In the presence of strong cross-section dependence (Tables 1 and 2), the P test shows severe size distortions, the W^* test exhibits good size properties, the $K_a^{\ast }$ and $K_b^{\ast }$ tests generally perform well but are, as expected, severely oversized when N = 50 and T = 20 because they are designed for cases of small N and large T values. The CIPS and S tests show size distortions for the case of negative serial correlation but perform reasonably well for other cases. The Z and W_m tests are slightly oversized but the $P_e^c$ test is conservative most of the time. ⁶

Table 1.

Size and power of tests: factor structure and no serial correlation

	N	T	P	Z	S	CIPS	$P_e^c$	$K_a^{\ast }$	$K_b^{\ast }$	Wm	W*
δ=0	20	20	0.239	0.078	0.034	0.036	0.033	0.047	0.027	0.081	0.065
		50	0.222	0.072	0.030	0.050	0.030	0.033	0.031	0.077	0.075
		100	0.236	0.071	0.034	0.045	0.040	0.029	0.032	0.074	0.080
	50	20	0.271	0.076	0.030	0.037	0.025	0.217	0.196	0.079	0.063
		50	0.292	0.071	0.022	0.042	0.019	0.101	0.080	0.073	0.073
		100	0.287	0.068	0.034	0.050	0.022	0.083	0.075	0.070	0.075
δ=0.1	20	20	0.228	0.084	0.043	0.032	0.024	0.033	0.026	0.093	0.067
		50	0.257	0.087	0.037	0.256	0.020	0.073	0.058	0.092	0.081
		100	0.280	0.094	0.070	0.621	0.024	0.135	0.142	0.100	0.084
	50	20	0.304	0.094	0.035	0.020	0.016	0.231	0.171	0.102	0.075
		50	0.311	0.080	0.037	0.312	0.020	0.147	0.133	0.085	0.083
		100	0.356	0.091	0.082	0.748	0.030	0.261	0.262	0.118	0.098
δ=0.5	20	20	0.254	0.110	0.040	0.008	0.058	0.035	0.030	0.120	0.065
		50	0.375	0.151	0.069	0.473	0.105	0.125	0.089	0.188	0.117
		100	0.634	0.345	0.191	0.980	0.135	0.330	0.346	0.408	0.328
	50	20	0.326	0.124	0.036	0.000	0.091	0.211	0.166	0.142	0.071
		50	0.456	0.152	0.063	0.583	0.158	0.264	0.224	0.184	0.145
		100	0.706	0.367	0.182	0.997	0.217	0.600	0.592	0.429	0.349
δ=0.9	20	20	0.296	0.117	0.051	0.018	0.088	0.060	0.039	0.120	0.092
		50	0.476	0.164	0.081	0.356	0.321	0.158	0.152	0.176	0.200
		100	0.832	0.460	0.275	0.958	0.637	0.463	0.461	0.467	0.532
	50	20	0.358	0.118	0.047	0.007	0.125	0.291	0.231	0.124	0.093
		50	0.556	0.177	0.083	0.466	0.456	0.467	0.449	0.189	0.210
		100	0.877	0.445	0.244	0.990	0.747	0.841	0.852	0.456	0.516

Note: Rejection rates of panel unit root tests at nominal level α=0.05. P is Maddala and Wu (1999)’s Fisher test, Z is Demetrescu et al. (2006)’s modified inverse normal test, S is Hanck (2008)’s Simes test, CIPS is Pesaran (2007)’s cross-sectionally augmented IPS test, $P_e^c$ is Bai and Ng (2004)’s pooled test statistic on idiosyncratic components, $K_a^{\ast }$ is Chang (2004)’s $K_{OT}^{\ast }$ test, $K_b^{\ast }$ is Palm et al. (2008)’s modified $K_{OT}^{\ast }$ test, W_m is the modified TPM and W^* is the bootstrap TPM.

Table 2.

Size and power of tests: factor structure with serial correlation

	N	T	P	Z	S	CIPS	$P_e^c$	$K_a^{\ast }$	$K_b^{\ast }$	Wm	W*
			(a) Positive AR serial correlation
δ=0	20	20	0.184	0.095	0.030	0.030	0.050	0.044	0.025	0.089	0.051
		50	0.197	0.083	0.030	0.023	0.032	0.056	0.051	0.084	0.063
		100	0.173	0.073	0.028	0.033	0.032	0.082	0.068	0.072	0.060
	50	20	0.265	0.100	0.038	0.025	0.033	0.226	0.181	0.100	0.054
		50	0.242	0.081	0.026	0.040	0.019	0.173	0.161	0.087	0.055
		100	0.241	0.076	0.025	0.045	0.027	0.200	0.192	0.077	0.068
δ=0.1	20	20	0.198	0.095	0.031	0.026	0.042	0.044	0.023	0.093	0.055
		50	0.222	0.086	0.035	0.089	0.026	0.078	0.063	0.089	0.063
		100	0.239	0.094	0.062	0.301	0.038	0.216	0.196	0.108	0.075
	50	20	0.259	0.096	0.034	0.019	0.027	0.194	0.164	0.097	0.054
		50	0.289	0.102	0.036	0.103	0.021	0.237	0.213	0.106	0.076
		100	0.325	0.102	0.055	0.401	0.040	0.328	0.414	0.112	0.076
δ=0.5	20	20	0.229	0.116	0.044	0.007	0.072	0.062	0.035	0.136	0.059
		50	0.320	0.130	0.050	0.272	0.113	0.196	0.149	0.158	0.111
		100	0.558	0.305	0.134	0.882	0.163	0.491	0.454	0.363	0.260
	50	20	0.274	0.106	0.032	0.002	0.101	0.200	0.146	0.125	0.053
		50	0.404	0.146	0.048	0.257	0.160	0.385	0.364	0.176	0.103
		100	0.654	0.334	0.120	0.963	0.249	0.786	0.783	0.411	0.303
δ=0.9	20	20	0.246	0.103	0.037	0.017	0.120	0.073	0.060	0.108	0.064
		50	0.428	0.158	0.060	0.152	0.386	0.281	0.215	0.165	0.166
		100	0.761	0.377	0.187	0.818	0.738	0.595	0.590	0.385	0.434
	50	20	0.327	0.129	0.034	0.013	0.158	0.244	0.201	0.137	0.089
		50	0.505	0.154	0.052	0.272	0.563	0.588	0.538	0.167	0.175
		100	0.829	0.393	0.169	0.967	0.855	0.935	0.934	0.410	0.455
			(b) Negative MA serial correlation
δ=0	20	20	0.280	0.087	0.063	0.134	0.039	0.030	0.018	0.094	0.077
		50	0.278	0.090	0.079	0.200	0.034	0.015	0.012	0.093	0.091
		100	0.276	0.082	0.096	0.213	0.035	0.015	0.014	0.081	0.088
	50	20	0.315	0.078	0.057	0.151	0.023	0.205	0.211	0.084	0.083
		50	0.344	0.085	0.097	0.256	0.021	0.053	0.042	0.089	0.087
		100	0.341	0.074	0.111	0.312	0.032	0.034	0.044	0.078	0.095
δ=0.1	20	20	0.281	0.092	0.055	0.080	0.027	0.031	0.020	0.098	0.079
		50	0.315	0.100	0.107	0.488	0.025	0.039	0.026	0.108	0.105
		100	0.337	0.110	0.156	0.819	0.026	0.082	0.090	0.135	0.106
	50	20	0.325	0.082	0.062	0.094	0.021	0.201	0.189	0.086	0.075
		50	0.352	0.080	0.118	0.589	0.027	0.092	0.079	0.094	0.082
		100	0.386	0.103	0.198	0.938	0.034	0.168	0.152	0.133	0.112
δ=0.5	20	20	0.305	0.110	0.070	0.010	0.065	0.047	0.025	0.129	0.089
		50	0.428	0.180	0.154	0.607	0.121	0.084	0.065	0.211	0.161
		100	0.682	0.405	0.324	0.995	0.144	0.262	0.266	0.468	0.412
	50	20	0.374	0.136	0.071	0.004	0.105	0.178	0.157	0.155	0.092
		50	0.502	0.184	0.173	0.757	0.170	0.176	0.155	0.225	0.175
		100	0.744	0.408	0.399	1.000	0.191	0.438	0.443	0.481	0.404
δ=0.9	20	20	0.338	0.124	0.075	0.035	0.065	0.046	0.041	0.132	0.103
		50	0.563	0.218	0.176	0.509	0.253	0.102	0.095	0.221	0.256
		100	0.837	0.525	0.438	0.978	0.525	0.344	0.362	0.519	0.554
	50	20	0.391	0.119	0.081	0.031	0.102	0.269	0.233	0.132	0.113
		50	0.605	0.218	0.236	0.715	0.358	0.362	0.316	0.229	0.236
		100	0.885	0.543	0.532	0.996	0.584	0.676	0.682	0.541	0.568

Table 3 reports the results with spatial dependence. Under SAR specification, the CIPS, $P_e^c$ and W_m tests exhibit size distortions, mainly because spatial correlation is typically weak and not captured by a common factor or constant correlation assumption. Additionally, ignoring such a weak correlation leads to over-rejection, as shown by the result from the P test. The $K_a^{\ast }$ and $K_b^{\ast }$ tests are severely undersized. The W^* test is undersized for small T values, but the size distortion reduces as T becomes large. The S test performs reasonably well. Under SMA specification, while all bootstrap tests suffer from downward size distortions, the P, CIPS and W_m tests are slightly oversized.

Table 3.

Size and power of tests under spatial dependence

	N	T	P	Z	S	CIPS	$P_e^c$	$K_a^{\ast }$	$K_b^{\ast }$	Wm	W*
			(a) Spatial AR dependence
δ=0	20	20	0.126	0.076	0.049	0.115	0.092	0.009	0.006	0.092	0.021
		50	0.135	0.070	0.046	0.118	0.073	0.008	0.007	0.079	0.040
		100	0.118	0.071	0.044	0.138	0.082	0.004	0.003	0.067	0.054
	50	20	0.117	0.051	0.052	0.096	0.071	0.013	0.013	0.092	0.014
		50	0.129	0.056	0.058	0.109	0.062	0.002	0.002	0.080	0.037
		100	0.125	0.056	0.047	0.097	0.088	0.001	0.001	0.080	0.048
δ=0.1	20	20	0.124	0.078	0.056	0.120	0.094	0.011	0.011	0.114	0.027
		50	0.147	0.092	0.060	0.131	0.091	0.011	0.011	0.130	0.052
		100	0.228	0.134	0.104	0.188	0.127	0.014	0.014	0.196	0.088
	50	20	0.145	0.067	0.053	0.081	0.076	0.011	0.012	0.137	0.021
		50	0.181	0.083	0.051	0.124	0.102	0.006	0.006	0.157	0.064
		100	0.318	0.154	0.106	0.172	0.173	0.007	0.006	0.308	0.166
δ=0.5	20	20	0.174	0.098	0.061	0.132	0.148	0.014	0.011	0.138	0.042
		50	0.361	0.195	0.086	0.240	0.265	0.035	0.034	0.227	0.156
		100	0.781	0.548	0.266	0.520	0.506	0.141	0.116	0.611	0.579
	50	20	0.208	0.095	0.071	0.105	0.155	0.017	0.018	0.165	0.036
		50	0.564	0.275	0.103	0.266	0.456	0.032	0.030	0.317	0.244
		100	0.970	0.822	0.314	0.667	0.849	0.186	0.210	0.847	0.891
δ=0.9	20	20	0.225	0.129	0.064	0.153	0.201	0.025	0.023	0.161	0.051
		50	0.621	0.270	0.133	0.454	0.583	0.088	0.071	0.273	0.284
		100	0.984	0.721	0.434	0.922	0.957	0.474	0.415	0.721	0.887
	50	20	0.296	0.140	0.070	0.142	0.290	0.021	0.020	0.182	0.058
		50	0.880	0.321	0.122	0.544	0.878	0.109	0.116	0.319	0.487
		100	1.000	0.879	0.478	0.995	1.000	0.748	0.752	0.872	0.996
			(b) Spatial MA dependence
δ=0	20	20	0.070	0.049	0.044	0.072	0.060	0.024	0.023	0.070	0.017
		50	0.075	0.050	0.052	0.089	0.046	0.013	0.015	0.074	0.023
		100	0.077	0.051	0.042	0.088	0.053	0.019	0.020	0.074	0.041
	50	20	0.081	0.039	0.058	0.047	0.044	0.024	0.024	0.089	0.009
		50	0.078	0.036	0.043	0.062	0.035	0.012	0.010	0.071	0.017
		100	0.076	0.031	0.048	0.060	0.046	0.012	0.013	0.074	0.038
δ=0.1	20	20	0.080	0.054	0.050	0.066	0.055	0.025	0.021	0.102	0.024
		50	0.101	0.062	0.053	0.077	0.077	0.029	0.034	0.111	0.034
		100	0.185	0.121	0.097	0.132	0.125	0.056	0.058	0.212	0.107
	50	20	0.076	0.034	0.062	0.068	0.045	0.032	0.025	0.109	0.014
		50	0.133	0.058	0.050	0.085	0.088	0.029	0.025	0.146	0.033
		100	0.281	0.144	0.118	0.127	0.197	0.069	0.071	0.298	0.179
δ=0.5	20	20	0.125	0.078	0.055	0.082	0.101	0.031	0.038	0.121	0.032
		50	0.346	0.221	0.083	0.209	0.283	0.127	0.153	0.243	0.155
		100	0.862	0.668	0.280	0.525	0.667	0.568	0.573	0.700	0.689
	50	20	0.155	0.076	0.050	0.077	0.139	0.024	0.042	0.150	0.016
		50	0.600	0.330	0.098	0.239	0.576	0.213	0.215	0.354	0.232
		100	0.995	0.910	0.350	0.720	0.987	0.890	0.898	0.918	0.955
δ=0.9	20	20	0.187	0.118	0.067	0.109	0.177	0.055	0.054	0.151	0.048
		50	0.652	0.278	0.114	0.399	0.653	0.378	0.367	0.257	0.326
		100	0.999	0.801	0.498	0.953	0.990	0.952	0.968	0.802	0.946
	50	20	0.290	0.150	0.079	0.106	0.316	0.040	0.038	0.198	0.052
		50	0.942	0.344	0.130	0.535	0.956	0.572	0.548	0.320	0.544
		100	1.000	0.930	0.543	1.000	1.000	1.000	1.000	0.930	1.000

In general, the power of all tests increases when T increases and when N increases as long as δ is fixed, which justifies the use of panel data in unit root tests. All tests become more powerful when δ increases, which is consistent with the findings in Karlsson and Löthgren (2000).

When comparing the different tests under varying scenarios, there is no dominant test. For example, the S test outperforms the others when very few series in the panel are stationary, the CIPS test performs the best under the factor structure, the $P_e^c$ test is preferable under spatial dependence, and the $K_a^{\ast }$ and $K_b^{\ast }$ tests behave similarly and well under factor structure but not under SAR specification. The W_m, Z and W^* tests have comparable power and all perform well under a factor structure with negative serial correlation and spatial dependence.⁷ We disregard the P test due to its immense size distortions under cross-section dependence.

4 Empirical Application

In the analysis of the term structure of macroeconomic forecasts, Lahiri and Sheng (2008) proposed a Bayesian learning model, implying that forecast precision, if perceived properly, should contain a unit root. We test this implication directly by using density forecasts for inflation and real GDP.

Following the terminology in Lahiri and Sheng (2008), the precision of individual i’s belief is evolved according to the following equation:

$$a_{\mathit{ith}}=a_{it,h+1}+b_{\mathit{ith}},$$ (12)

where a_ith is the precision of individual i’s posterior belief in predicting the variable for the target year t and h quarters ahead to the end of the target year, and a_it,h+1 is the precision of his prior belief at h +1 quarters ahead to the end of the target year t. Here b_ith is individual i’s perceived quality of public information, which measures the shock to his precision updating process.

The data in this study are taken from Survey of Professional Forecasters (SPF), which uniquely includes density forecasts for inflation and real GDP. For several reasons as stated in Engelberg et al. (2009), we restrict attention to data collected from the first quarter of 1992 to the second quarter of 2009. Due to the design of the survey, the actual horizons for these forecasts are approximately from 8 quarters to 1 quarter for a target year. This fixed-target scheme enables us to study the evolution of forecast precision over horizons. For the purpose of estimation, we eliminate observations by infrequent respondents, and focus on the “regular” respondents who participated in at least 50 percent of the forecast periods. This leaves us with 24 individuals, whose identification numbers are listed in Table 4.⁸ The precision a_ith is calculated as the reciprocal of the variance of the density forecast reported by individual i.⁹

Table 4.

Panel unit root tests of forecast precision

ID	Inflation			Real GDP
	DF statistics	p-value	Simes criterion	DF statistics	p-value	Simes criterion
20	−0.94	0.298	0.019	2.12	0.989	0.027
65	−2.72	0.009	0.008	6.97	1.000	0.038
84	−0.16	0.616	0.025	−1.15	0.219	0.008
99	3.62	1.000	0.044	2.81	0.998	0.033
407	−3.91	0.001	0.002	2.82	0.998	0.035
411	−2.60	0.012	0.010	3.30	1.000	0.040
420	−1.84	0.063	0.015	−0.56	0.460	0.015
421	0.55	0.827	0.029	−3.90	0.001	0.002
426	−1.21	0.201	0.017	−0.99	0.278	0.010
428	1.72	0.975	0.033	−2.25	0.027	0.004
431	2.59	0.996	0.042	−0.07	0.647	0.019
433	2.32	0.993	0.038	−0.35	0.547	0.017
439	4.01	1.000	0.046	1.12	0.926	0.025
446	−0.42	0.517	0.021	4.87	1.000	0.042
456	−2.43	0.018	0.013	5.21	1.000	0.044
463	3.55	1.000	0.048	4.20	1.000	0.046
472	2.18	0.990	0.035	0.95	0.903	0.023
483	2.46	0.995	0.040	2.25	0.992	0.029
484	3.31	1.000	0.050	5.28	1.000	0.048
504	1.55	0.965	0.031	0.93	0.900	0.021
507	0.00	0.670	0.027	−1.80	0.069	0.006
508	−3.19	0.003	0.004	−0.83	0.343	0.013
510	−2.73	0.009	0.006	4.64	1.000	0.050
512	−0.17	0.614	0.023	2.37	0.993	0.031
P		0.004			0.796
Z		1.000			1.000
Wm		0.000			0.023

Note: The DF statistics are based on univariate AR(1) specification in the level of the variable without an intercept. The corresponding p-values are obtained using the response surfaces estimated in MacKinnon (1996). Simes criterion is calculated as iα/N based on ordered p-values for i = 1, …, N. P is Maddala and Wu (1999)’s Fisher test, Z is Demetrescu et al. (2006)’s modified inverse normal test and W_m is the modified TPM. All statistics are calculated based on the same sample period, namely 1992:Q1-2009:Q2, using the density forecasts from Survey of Professional Forecasters. The significance level α is set at 0.05.

We first estimate individual DF regressions and then compute the pairwise cross-section correlation coefficient of the residuals. In our sample the average of these correlation coefficients is estimated to be 0.07 for inflation and 0.09 for real GDP. The CD statistics, 9.41 for inflation and 11.70 for real GDP, strongly reject the null of no cross-section dependence for both variables.

Now we turn to panel unit root tests that account for this positive cross-section dependence.¹⁰ The joint null and alternative hypotheses are specified in (3) and (4). For inflation forecasts, the S, P and W_m tests reject the joint null hypothesis of non-stationarity in forecasters’ precision updating process at the 5% significance level, but the Z test fails to reject the null. As for real GDP forecasts, the S and W_m tests show strong evidence of rejection, but the P and Z tests do not reject. To understand the mixed evidence against the null, recall that the Z test uses all p-values and tends to lose power when some p-values are very large. In this example, about 40% of the p-values are close to 1 for inflation and 60% for real GDP. In contrast, by truncating these large p-values are removed, increasing the power of W_m. The S test is also powerful in this case, since there are some very small and reinforcing p-values in the panel. Thus, the evidence from panel data analysis shows that in predicting real GDP and inflation, some professional forecasters do not update their forecast precision in an optimal Bayesian way. Using the approaches of Hanck (2009) or Moon and Perron (2011) may give insight to which forecasters do not behave optimally.

5 Conclusion

In this paper we propose two panel unit root tests: the modified TPM, W_m, and the residual-based bootstrap TPM, W^*. The W_m test allows for an unbalanced panel and the W^* test is robust to general forms of cross-section dependence in the panel. We conduct a systematic comparison of the proposed tests with other commonly used panel unit root tests. Evidence from Monte Carlo simulations shows that both tests deliver good power with moderate and large T values. Although the W_m test is slightly oversized, the W^* test yields good empirical size. To illustrate the usefulness of the proposed tests, we apply them to a panel of real GDP and inflation density forecasts. The resulting evidence indicates that some professional forecasters do not update their forecast precision in an optimal Beyesian way.

Our approach can be extended in a number of directions. One obvious generalization is to incorporate weights, thus allowing tests of more precision to play a larger role. Another worthwhile extension would be to develop an adaptive TPM that optimizes the selection of the truncation point among a set of candidates.