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Econometrics of Panel Data Jakub Mućk Meeting # 9 Jakub Mućk Econometrics of Panel Data Meeting # 9 1 / 22

Outline 1 Time series analysis Stationarity Unit Root Tests for Nonstationarity 2 Panel Unit Root Tests LLC test IPS test Fisher-type test CADF test Chang test Jakub Mućk Econometrics of Panel Data Time series analysis Meeting # 9 2 / 22

Stationarity definition Weak stationarity 1 Constant mean function of y t : E(y t ) = µ. (1) 2 Bounded variance of the y t Var(y t ) = E(y t µ) 2 = σ 2 <. (2) 3 Constant autocovariance function of y t : Cov(y t, y t+k ) = E(y t µ)(y t+k µ) = λ k. (3) If the series is non stationary it has an unit root. Jakub Mućk Econometrics of Panel Data Time series analysis Meeting # 9 3 / 22

Examples of stochastic processes White noise: is stationary, where ε t N(0, σ 2 ) and cov(ε t, ε s ) = 0 fort s: Random walk y t = ε t (4) y t = y t 1 + ε t (5) is non-stationary, because its variance cannot be limited. AR(1) process y t = ρy t 1 + ε t (6) is stationary only if the ρ is less in modulus than unity, i.e., ρ < 1. Jakub Mućk Econometrics of Panel Data Time series analysis Meeting # 9 4 / 22

Empirical examples Macrodata for the US economy: Unemployment rate (Ut); Logged Real Gross Domestic Product (ln GDPt). Time span: 1948Q1 2016Q4. Jakub Mućk Econometrics of Panel Data Time series analysis Meeting # 9 5 / 22

Unemployment rate U t 2 4 6 8 10 1950q1 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 2020q1 Jakub Mućk Econometrics of Panel Data Time series analysis Meeting # 9 6 / 22

Logged Real Gross Domestic Product ln GDP t 7.5 8 8.5 9 9.5 10 1950q1 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 2020q1 Jakub Mućk Econometrics of Panel Data Time series analysis Meeting # 9 7 / 22

First difference of GDP ln GDP t -.04 -.02 0.02.04 1950q1 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 2020q1 Jakub Mućk Econometrics of Panel Data Time series analysis Meeting # 9 8 / 22

Dickey Fuller test (basic) I The general assumption: y t is generated by AR(1): y t = ρy t 1 + ε t (7) The general idea is to test whether ρ is equal or significantly less than one. The null is that there is unit root (y t is nonstationary) Estimating ρ in equation (7) and calculating t-statistics might lead to completely meaningless result (spurious regression). Therefore, y t should be differenced: where ε t N (0, σ 2 ) and γ = (1 ρ). y t = (ρ 1) y t 1 + ε t = γy t 1 + ε t (8) The null (y t is stationary) and alternative (y t is stationary) H 0 : ρ = 1 H 0 : γ = 0 H 1 : ρ < 1 H 1 : γ < 0 (9) Jakub Mućk Econometrics of Panel Data Time series analysis Meeting # 9 9 / 22

Dickey Fuller test (basic) II To check stationary we estimate equation (7) and calculate t-statistic. Here, we cannot use t distribution because the calculated t statistic is not t distributed. Therefore we use τ statistics which equals t-statistics. Critical values for τ statistic are computed from numerical distributions. The null is rejected when τ is below its critical value. One might include in regression (7) deterministic components: constant or time trend. The null and alternative will be the same. Jakub Mućk Econometrics of Panel Data Time series analysis Meeting # 9 10 / 22

Augmented Dickey Fuller test To avoid the danger of autocorrelation of error test we might extend test regression by autoregression part of higher order: y t = γy t 1 + P α s y t s + ε t (10) The null and the alternative are the same as in the basic version. i=1 H 0 : γ = 0 H 1 : γ < 0 (11) In practice, including autoregressive part in DF regression is very often approach. The ADF test can be extended by deterministic component: constant or trend. Jakub Mućk Econometrics of Panel Data Time series analysis Meeting # 9 11 / 22

Critical values The critical values differ from t-student distribution. Table: Critical values for DF test Model 1% 5% 10% y t = γy t 1 ε t -2.56-1.94-1.62 y t = α + γy t 1 ε t -3.43-2.86-2.57 y t = α + δt + γy t 1 ε t -3.96-3.41-3.13 Standard critical values -2.33-1.65-1.28 Note: critical values are taken from Davidson and MacKinnon (1993) Jakub Mućk Econometrics of Panel Data Time series analysis Meeting # 9 12 / 22

Panel Unit Root Tests The panel unit root tests allow to investigate mean-reversion (stationarity) in the group (panel) of series. The baseline framework is the ADF regression for panel data: where γ i = ρ i 1. y it = γ i y it 1 + P α j y it j + ε it, (12) j=1 The typical null hypothesis for testing non stationarity H 0 : γ i = 0 (or ρ i = 1). (13) The alternative hypothesis is not common for the panel unit root test that based on the ADF regression: 1 H 1 : γ i = γ < 0 (or ρ i = ρ < 1) for all panels. 2 H 1 : γ i < 0 (or ρ i < 1) for some panels. Jakub Mućk Econometrics of Panel Data Panel Unit Root Tests Meeting # 9 14 / 22

LLC test I Levin, Lin and Chu (2002, henceforth LLC) propose the test in which homogeneity of the rho is assumed: i H 0 : γ i = 0 (ρ i = 1), i H 1 : γ i < 0 (ρ i < 1). The LLC test based on the pooled fixed-effects regression that allow to estimates t-ratio of γ parameter. General procedure: 1 The ADF regressions for each cross-section: y it = γ iy it 1 + P α j y it j + ε it, (14) and calculate the error variance ˆσ ε,i 2 For each cross-section use two auxiliary regression to get orthogonalized residuals: j=1 Jakub Mućk Econometrics of Panel Data Panel Unit Root Tests Meeting # 9 15 / 22

LLC test II 1 ẽ it = ê it /ˆσ ε,i where ê it are the residuals from regression of y it on its lagged values. 2 ṽ it 1 = ˆv it /ˆσ ε,i where ˆv it 1 are the residuals from regression of y it 1 on y it j where j {1,..., P}. 3 Estimates the ratio of the long-run to short-run variance. 4 Consider the pooled regression: ẽ it = γṽ it 1 + ν it (15) to get unadjusted t statistic, i.e., t = ˆγ/se(ˆγ). 5 Calculate the adjusted statistics t that takes into account the mean and standard deviation adjustment. Under the null hypothesis the t N (0, 1) as N T /T 0. Jakub Mućk Econometrics of Panel Data Panel Unit Root Tests Meeting # 9 16 / 22

IPS test The IPS test (Im, Pesaran and Shin, 2003) is less restrictive than the LLC test. Namely, it is assumed that i H 0 : γ i = 0 (ρ i = 1), i H 1 : γ i < 0 (ρ i < 1). The test statistic t is based on averaging over units the individual ADF statistics t i : t = 1 N t i. (16) N Alternatively, the Z t statistics can be used. the Z t statistics is the standardized version. It T followed by N sequentially then Z t N (0, 1). t=1 Jakub Mućk Econometrics of Panel Data Panel Unit Root Tests Meeting # 9 17 / 22

Fisher-type test I Maddala and Wu (1999) and Choi (2001) propose a test that combines the p-values of the individuals statistics. The alternative hypothesis refers to the heterogeneous case: i H 0 : γ i = 0 (ρ i = 1), i H 1 : γ i < 0 (ρ i < 1). The combined test statistics is following: π = 2 N ln(π i ), (17) i=1 where π i is the p-value from ith cross-section. The test statistics is χ 2 distributed with 2N degrees of freedom. Jakub Mućk Econometrics of Panel Data Panel Unit Root Tests Meeting # 9 18 / 22

Fisher-type test II If N is large then the transformed version of π: P = 1 N i = 1 N [ln(π i ) + 1]. (18) The P statistics have a standard distribution when T, N, sequentially. Alternatively, the inverse normal distribution can be used: Z INV = 1 N i = 1 N Φ 1 (π i ) (19) where Φ() is the conditional distribution function of the standard normal distribution. Jakub Mućk Econometrics of Panel Data Panel Unit Root Tests Meeting # 9 19 / 22

Panel Unit Root Tests Pesaran (2007) proposes the cross-sectionally augmented Dickey Fuller (CADF) test. The extended ADF regression: y it = γ i y it 1 + P Q α j y it j + δ 0 ȳ t 1 + δ j ȳ t j + ε it, (20) j=1 where ȳ t is the cross-sectional average. The CIPS statistics is calculated as the unweighted average from the individuals CADF statistics (t statistics on y it 1 ): CIPS = 1 N j=0 N CADF i. (21) Akin to the IPS test, it is possible to calculate the Z t statistics which have a standard normal distribution. i=1 Jakub Mućk Econometrics of Panel Data Panel Unit Root Tests Meeting # 9 20 / 22

Chang test Chang (2002) proposes alternative method to deal with the cross-sectional dependence. This approach bases on the nonlinear IV methodology. The general idea is to use nonlinear transformation of the lagged dependent variable as its instrument: z it = F(y it 1 ), (22) where F() is called instrument generating function (IGF). As in the previous cases, the test S statistics bases on aggregating and normalizing the t-statistics from the cross-sectional (IV) regression. Under the null, the S is standard normally distributed. The IGF function should be integrable and satisfy the following condition: for instance: xf(x)dx 0, (23) F(x) exp( c x ) (24) where c is some factor. Chang and Song (2008) postulate to extend the ADF regression by the appropriate covariate (e.g. y jt 1 for other units or cross sectional averages.) Jakub Mućk Econometrics of Panel Data Panel Unit Root Tests Meeting # 9 21 / 22

Summary Test Heterogeneity in ρ Unbalanced cross-sectional independence LLC common ρ balanced IPS heterogeneous ρ balanced/unbalanced Fisher-type heterogeneous ρ balanced/unbalanced cross-sectional dependence CADF heterogeneous ρ balanced/unbalanced Chang heterogeneous ρ balanced/unbalanced Jakub Mućk Econometrics of Panel Data Panel Unit Root Tests Meeting # 9 22 / 22