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1、1 Unobserved heterogeneity in panel time series models Jerry Coakleya, Ana-Maria Fuertesb, Ron Smithc aDepartment of Accounting, Finance and Management, University of Essex, Colchester CO4 3SQ, UK. bFaculty of Finance, Cass Business School, 106 Bunhill Row, London EC1Y 8TZ, UK. cDepartment of Econom
2、ics, Birkbeck College, Malet Street, London WC1E 7HX, UK. October 2004 Abstract Recently, the large T panel literature has emphasized unobserved, time-varying heterogeneity that may stem from omitted common variables or global shocks that a?ect each individual unit di?erently. These latent common fa
3、ctors induce cross-section dependence and may lead to inconsistent regression coe?cient estimates if they are correlated with the explanatory variables. Moreover, if the process underlying these factors is nonstationary, the individual regressions will be spurious but pooling or averaging across ind
4、ividual estimates still permits consistent estimation of a long-run coe?cient. The need to tackle both error cross-section dependence and persistent autocorrelation is motivated by the evidence of their pervasiveness found in three well-known, international ?nance and macroeconomic examples. A range
5、 of estimators is surveyed and their ?nite-sample properties are examined by means of Monte Carlo experiments. These reveal that a mean group version of the common-correlated-e?ects estimator stands out as the most robust since it is the preferred choice in rather general (non) stationary settings w
6、here regressors and errors share common factors and their factor loadings are possibly dependent. Other approaches which perform reasonably well include the two-way ?xed e?ects, demeaned mean group and between estimators but they are less e?cient than the common-correlated-e?ects estimator. Keywords
7、: Factor analysis; global shocks; latent variables JEL Classi?cation: C32; F31 1 Introduction Panel or longitudinal data which have observations on cross-section units i = 1; 2; :; N; such as individuals, ?rms or countries, over time periods t = 1; 2; :; T enable one to model a variety of forms of u
8、nobserved heterogeneity in regression models. The standard panel literature, developed Corresponding author: Tel. +44-01206-872455; fax: +44-01206-873429. E-mail address: jcoak- leyessex.ac.uk (J. Coakley). 2 for cases where N is large and T is small, emphasizes unit-speci?c heterogeneity such as un
9、observed ability in earnings equations. When T is large, one can allow for such unit-speci?c heterogeneity by estimating a separate time-series equation for each unit. Recent years have witnessed increasing interest in panel data models with unobserved time-varying heterogeneity induced by common sh
10、ocks that in?uence all units, perhaps to di?erent degrees. This is particularly important in international ?nance and macroeconomics where long runs of data are available for many countries, each of which may be subject to global shocks. Such heterogeneity will introduce cross-section dependence or
11、correlation between the errors of di?erent units and will render the conventional estimators inconsistent if the global shocks are correlated with the regressors. It is also quite plausible that these unobserved factors, such as technology shocks in a production function or ?nancial innovation in a
12、money demand function, may need ?rst di?erencing to achieve stationarity. Such I(1) shocks cause the variables not to cointegrate and the regression to be spurious, that is, the covariance between the I(1) error and the I(1) regressor does not go to zero even as T ! 1 and so the estimator does not c
13、onverge to the true parameter value but to a random variable. However, Phillips and Moon (1999, 2000) and Kao (1999) show that panels make it possible to obtain consistent estimators (as N ! 1) of a long-run average parameter even when each of the individual time-series regressions is spurious: The
14、averaging over N attenuates the noise in the individual estimators and thus facilitates a consistent estimator of the mean e?ect. In the panel time-series literature where both N and T are large, the usual approach has been either to ignore the possibility of cross-section dependence produced by tim
15、e-speci?c heterogeneity or deal with it by including period dummies or ?xed e?ects. But this assumes that the global shocks have identical e?ects on each unit which seems quite restrictive. When N is of the same order of magnitude or greater than T , the traditional SUR-GLS approach for dealing with
16、 cross-section de- pendence breaks down because the estimated contemporaneous variance-covariance matrix cannot be inverted. If T is only slightly greater than N, estimation is feasible but it will be unreliable. However, assuming cross-section independence seems restrictive for many applications in
17、 macro- economics and ?nance and neglecting it may be far from innocuous as has been clear in the purchasing power parity (PPP) debate (see O?Connell, 1998). Phillips and Sul (2003) note that pooling may provide little gain in precision over single-equation estimation if there is substantial cross-s
18、ection dependence. In addition, many theoretical results have been derived under the as- 3 t sumption of independence (Phillips and Moon, 2000). As Phillips and Moon (1999: p1092) put it ?.quite commonly in panel data theory, cross-section independence is assumed in part because of the di?culties of
19、 characterizing and modelling cross-section dependence.? In spatial econometrics, quite popular in urban economics and regional science, a natural way to model cross-section dependence is in terms of distance (see Baltagi, 2001). But for most economic problems there is no obvious distance measure. I
20、n recent years, characterizing cross- section dependence by means of a factor structure has attracted a lot of attention (Robertson and Symons, 1999; Bai and Ng, 2002; Coakley, Fuertes and Smith, 2002; Phillips and Sul, 2003; Moon and Perron, 2004; Pesaran 2004a). Accordingly, the disturbances are a
21、ssumed to contain one or more unobserved (latent) factors which may in?uence each unit di?erently. This paper examines the consequences of time-varying heterogeneity that arises from unob- served factors, which are possibly I(1) processes, and the relative e?ectiveness of various approaches in deali
22、ng with this phenomenon. The focus of the analysis is on estimation issues rather than inference. Section 2 provides an empirical illustration of the problems. It shows that three stan- dard bivariate economic relations involve substantial cross-section dependence and the residuals resemble I(1) ser
23、ies. Section 3 discusses a range of possible estimators. Since we want to make the paper accessible to a wide audience, we indicate the nature of the issues rather than provide formal proofs or derivations. Section 4 provides Monte Carlo evidence on the ?nite sample properties of these estimators un
24、der various data generation processes and Section 5 concludes. 2 Empirical illustrations We take three standard applications to assess the extent of the two problems, cross-section depen- dence and I(1) errors, and to help in designing our Monte Carlo experiments. The applications are PPP, the Fishe
25、r relationship and the Feldstein-Horioka (FH) puzzle. Each of them involves a simple bivariate linear relationship that should hold in the long run. Let sit be the logarithm of the nominal exchange rate and dit = pit p the log price di?erential between country i and the base country (the US) at peri
26、od t. According to PPP, exchange rates should re?ect price ?uctuations in the long-run so in the regression sit = i + idit + eit; (1) 4 the restriction i = 1 should hold. Boyd and Smith (1999) and Coakley, Flood, Fuertes and Taylor (2004) provide further discussion. Let ilit denote the annualized lo
27、ng-term nominal interest rate and it the log annual in?ation rate. Assuming Et( i;t+1) = it, the ex ante real interest rate is rlit = ilit it: The Fisher e?ect suggests that nominal interest rates fully re?ect in?ation expectations in the long-run. Thus in ilit = i + i it + eit; (2) the restriction
28、i = 1 should hold. Coakley, Fuertes and Wood (2004) discuss this in more detail. In both examples, one might expect common (across countries) factors to be present. These would include base country e?ects, oil price shocks and the long swings in the real dollar rate for PPP and movements in the worl
29、d real interest rate for the Fisher equation. Let Iit be the share of domestic ?xed investment in GDP and Sit the share of savings. In a world of free capital mobility, national saving would ?ow to the countries o?ering the highest returns and domestic investment would be ?nanced from global capital
30、 markets. Thus in Iit = i + iSit + eit; (3) i = 0 should hold. The puzzle is that Feldstein and Horioka (1980) found the average i for OECD countries to be close to unity, the expected value under no capital mobility. Coakley, Kulasi and Smith (1996, 1998) and Coakley, Fuertes and Spagnolo (2004) pr
31、ovide further discussion. The analysis for the PPP and Fisher equations is based on quarterly data for 18 countries (Aus- tralia, Austria, Belgium, Canada, Denmark, France, Germany, Ireland, Italy, Japan, Netherlands, New Zealand, Norway, South Africa, Sweden, Switzerland, UK and US) over the 1973Q1
32、-1998Q4 period. The panel dimensions for the PPP analysis are N = 17 (US is excluded) and T = 104 while those for the Fisher regression are N = 18 (US is included) and T = 100 (four observations are lost in calculating the annual in?ation series it = pit pi;t 4): Nominal exchange rates and prices ar
33、e scaled (1995=100) to remove the e?ect of units of measurement on the intercepts. Long-term interest rates are average yields to maturity on bellwether government bonds with residual matu- rities between 9 and 10 years. All the price indexes are CPI series except for Australia where the PPI is used
34、 due to data unavailability. The FH regression is based on quarterly national saving, domestic investment and GDP observations for 12 OECD economies (Australia, Canada, Finland, France, Italy, Japan, Netherlands, Norway, Spain, Switzerland, UK and US) over 1980Q1-2000Q4. 5 Table 1 gives various summ
35、ary statistics for the variables and two sets of residuals coming from individual OLS and from two-way ?xed e?ects (2FE). Both levels and ?rst di?erences are considered. The 2FE estimator imposes slope coe?cient (and error variance) homogeneity but allows for country e?ects i and time e?ects t: The
36、latter may pick up any common factor. Table 1 around here On the one hand, Table 1 reports the average (absolute) correlation as an indication of the degree of cross-section dependence ? Pesaran (2004b) proposes a test for cross-section dependence based on the average correlation of the residuals an
37、d compares it with the Breusch-Pagan (1980) test based on the average of the squared correlations. On the other hand, Table 1 reports the proportion of the variance accounted for by the ?rst two principal components (PCs), as an indication of how well a factor structure works, and the average ADF t
38、-statistic of Im, Pesaran and Shin (2003) IPS as an indication of the possibility of a unit root. The PCs are the linear combinations of the standardized time series that account for the maximal amount of the total variation. The eigenvectors of the relevant correlation matrix are the weights and th
39、e ordered eigenvalues over the cumulated eigenvalues give the variance proportions. The ?rst PC often has roughly equal weights and so it is close to the cross-section mean of the data for each time period. The average absolute correlations between OLS residuals are 0:67 for PPP, 0:55 for Fisher and
40、 0:26 for FH. Using the 2FE estimator reduces the average absolute correlation in the PPP and (somewhat in) Fisher but not the FH case. There is little di?erence between the average absolute correlation and the average correlation (except for FH) since the residuals are mainly positively correlated.
41、 This is not always the case for the variables. In particular, the log price di?erential has an average absolute correlation of 0.84 but an average correlation of only 0.06 because large positive and negative correlations cancel out. The ?rst PC accounts for 72% of the OLS residual variance in the P
42、PP case, 61% in Fisher and 29% in FH and similarly for the 2FE residuals. In the PPP and Fisher cases, the ?rst two factors explain about 80% of the total residual variation. The IPS test is designed for variables (not residuals) and it assumes cross-section independence. Therefore, the average ADF
43、statistics should be treated as descriptive rather than as formal tests. The fact that these statistics are rather small (around -2) suggests that a unit root is likely to be present in the disturbances for many of the countries. There is slightly more evidence for a 6 t u;it u;it ui t x;it x;it xi
44、ft ft f unit root in the residuals from 2FE than in those from individual OLS, which is the reverse of what one would expect if there was an I(1) factor that the time ?xed e?ects have removed. The ?rst-di?erenced series yield much larger (absolute) average ADF statistics, as expected, and lower cros
45、s-section correlations. However, the residual dependence is still quite marked in the PPP and Fisher cases. This analysis illustrates that both cross-section dependence and potentially I(1) errors are a pervasive feature of the levels regressions (1)-(3). 3 Alternative panel estimators 3.1 The model
46、 Suppose that the data generating process is a linear heterogeneous panel model yit = i + ixit + uit, i = 1; :; N; t = 1; :; T; (4) where the parameters are distributed randomly over units, i = + i and i = + i with i s iid(0; 2 ) and i s iid(0; 2 ), and independently of xit and uit. Such random coe?
47、cient models (RCM) are discussed by Hsiao (2003) and Hsiao and Pesaran (2004). The variables and disturbances may be I(1) or I(0): The cross-section and time dependence structure is given by uit = uiuit 1 + if + ; iid(0; 2 ); (5) xit = xixit 1 + if + i t + ; iid(0; 2 ); (6) where iid denotes indepen
48、dence across t and i. Both ft and t are latent common factors such that ft may in?uence both errors (loading i) and regressors (loading i) whereas t is regressor speci?c. If i 6= 0 and i 6= 0, the error and regressor in (4) are correlated. We assume that u;it and x;it are independently distributed.
49、The factors may be I(0) or I(1) processes such as ft = f ft 1 + ; iid(0; 2 ); (7) t = t 1 + t; t iid(0; 2 ); (8) where ft and t are independently distributed. We do not consider lagged dependent variables as regressors because this raises a variety of quite di?erent issues central to a distinct literature on panel unit root testing surveyed by Trapani (2004). The parameter of interest is the mean e?ect