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This book provides a comprehensive and unified treatment of finite sample statistics and econometrics, a field that has evolved in the last five.
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- Four essays in finite-sample econometrics
- Graduate econometrics I
- Four essays in finite-sample econometrics
In this case, reduces to. One can also determine by nonparametric kernel estimators, where the vector zi can also include xi. This is given by where K is the weight kernel and h is the window width around the point zi. Essentially is an estimator of by data points which are in the interval h around zi.
The kernel K is chosen so that it gives low weight to the data on z which are far away from zi and high weight to the data which are close to zi. Using this nonparametric estimator of , we can get the operational or feasible GLS estimator , which is known as the semiparametric GLS estimator, see Pagan and Ullah , chapter 5. This estimator is useful in practice since obtaining the operational GLS estimator is not available when is unknown.
Also see Linton , for Edgeworth expansions. The iterative ML estimation is obtained by using where H is the Hessian matrix of log L, also see Engle for a simpler four-step procedure. For the hypothesis testing work related to grouped heteroskedasticity, see Sharma and Giaccotto and Bekker But these do not change the approximate results given below. Magee et al. The readers can attempt to develop this result alternatively by using the result in Section 2.
Thus, the variance of is more compared to due to estimation of. The variance MSE approximation given above also follows 5. The results are consistent with the theoretical results discussed above. The Edgeworth expansion of the distribution of 5. For inferential robustness they explore bounds for critical values of these tests which will ensure that the conclusions based on the usual tests are not affected against a particular class of elliptical distributions. Finally, we note that the exact distributions of and DW, which are the ratio of quadratic forms in y or u, have been extensively studied in Koerts and Abrahamse and Farebrother , , King , and others.
Further, the test is relatively robust for moderate nonnormality and large n. Such a model is needed when we wish to analyze a set of demand equations of different commodities or a set of investment functions for various industries. The simulation study in Srivastava and Maekawa suggests that the biases are generally small and insensitive to a wide range of asymmetric disturbances. But if the disturbances are symmetric, then the two estimators have the identical MSE matrices. The simulation results in some special cases of two equations in Srivastava and Maekawa suggest that the approximate results are quite accurate for small samples and robust to departures from normality of errors.
In general, relying on the normality of errors may give misleading results when the distributions are in fact nonnormal.
This model is useful for analyzing migration, labor force participation, decision of participating in a program or organization union , among others. Therefore, where ei is the error term. An extension of the above limited dependent variable is the sample selection model. So the LS regression of yi on xi will be asymptotically biased and inconsistent.
This is a model which adjusts for the sample selection bias. In many applied studies, it has been found that these two situations of large standard errors are frequently found. These results are consistent with the analytical results on the bias above. The analytical MSE results can also be written and analyzed from the results in Chapter 2.
Such a model is known as a panel data model, and such models are used to study the behavior of the cross-sections of groups, regions, countries, and households over time. There is an extensive literature on the theory and applications of these models; see Hsiao and Baltagi The possibility of observing variations across both cross-section and time points creates several interesting opportunities in applied econometrics; see Baltagi and Deaton for the advantages of panel data models.
A basic difference between the panel data models and the traditional models is the presence of changing varying intercepts across cross-sections and across time. There are two kinds of panel data models. In this case, the model is estimated by the LS procedure which provides the best linear unbiased estimator under the Gauss—Markov assumptions.
Second is the random effect error components model in which the heterogeneity parameters are treated as random error along with the equational random error. Such a model is estimated by the GLS method, the small sample properties of which will be discussed below. Second, an unbiased estimator is the between estimator represented by regression :. This is given by doing LS in the following which gives where transforms the data grouped by cross-sectional units into time averages for each unit.
Also, from Lehmann , theorem 1. But this result does not hold for the FGLS estimator. Thus, the total sample size does not help in the accuracy of the asymptotic moment approximation and that convergence is probably slower.
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This is given by where The following remarks follow from this result. This case implies or , that is, the random effect errors become the usual errors. The approximate bias expression in 5.
The result for approximate MSE can be similarly obtained. For the bias analysis in the nonlinear panel models, see Hahn and Newey This page intentionally left blank 6 Dynamic Time Series Model 6. Based on these results the bias and mean squared error MSE of the one period and s-periods ahead forecast errors have been analyzed.
Since unit root and cointegration play an important role in the modern time series, these cases are also covered in this chapter. Using recursive substitutions we can write 6. Further if y0 is stochastic then it is replaced by E y0 in 6. If then. The model in 6. The exact distribution of can be written by using the Imhof result on the distribution of the quadratic forms, see Chapter 3. An extensive numerical study of unit root distributions was done by Evans and Savin , , which warned that the conventional tests are not good detectors of unit root behavior since they have low powers in typical sample sizes.
These distributions can be used to test for unit root hypothesis. Phillips a,b,c extended White's distributional results using ratios of functions of Wiener processes, also see Abadir for the exact distribution. In addition to the work on the exact distribution, several authors have attempted to obtain approximations to the distribution of. Phillips a, , Satchell , Tsui and Ali , Wang , and Lieberman developed the Edgeworth expansion, Cornish—Fisher type expansion, and saddle point approximations of the distribution for the case of stationary AR 1 model.
Phillips a , Lieberman b , and Nabeya and Taneka have looked at the limiting distribution of under a near-integrated random process, also see Park This uniform expansion, however, provides a mixed result on its accuracy.
Now we turn to the works on the properties of forecasts. Malinvaud showed that if y0 is a mean zero symmetric random variable then the forecast bias of LS is zero. Phillips pointed out that this is an unconditional forecast error and that what is far more interesting is the forecast bias conditioned on known values of the dependent variable, such as yn.
Now we look into the results for the AR 1 with an intercept. From 6.
Further, assuming where c and C are given in 6. Thus the exact. This compares with the results of Malinvaud , Fuller and Hasza , and Dufour Similarly, at follows from 6. We can also notice from comparing Tables 6.
Four essays in finite-sample econometrics
It is also interesting to note that again using numerical methods and so. In general we observe from Tables 6. For the one period ahead forecast, see this analysis in Mercurio Thus, where Further,. The derivatives are straightforward and they are left as an exercise. As before the bias of the one period ahead forecast is can be obtained from 2. In fact O n bias approximations are essentially the same whether or not we condition on y0, since. It is easy to note that and. From these results, it appears no one approximation method dominates over all parameter values.
Graduate econometrics I
For larger values of sigma and larger values of alpha, the large-n approximation does the best. Then from 2. Abadir has looked into the accuracy of the asymptotic expansion with the exact results, also see Evans and Savin His results suggest that the approximate bias of is an overestimate of less than 0.
First, the exact distribution of , which is the ratio of quadratic forms, can be analyzed by the Imhof result in Chapter 3, see the contributions by Phillips a, , Tsui and Ali , and Evans and Savin However, if yn is an outlier the skewness of the conditional distribution is less marked. The approximation to the t-ratio statistic , where and is based on the LS residuals, is given by see Phillips a. Also see Phillips for the saddle point approximation. Given the approximate distributions one can also obtain the bias and MSE of. Then the exact moments of given in 2. Further where with The evaluations of and are as given in 6.
Four essays in finite-sample econometrics
The Laplace approximations of and are as given in 6. The exact and approximate moments of the one period ahead forecast error can then be obtained in the same way as in 6. The exact moments are then obtained by using the results of Sawa given in Chapter 2. These were evaluated numerically over the space of values of parameters involved in the moments. The bias is generally robust. Overall, while the bias is relatively insensitive to skewness and kurtosis the MSE is relatively sensitive to both skewness and kurtosis and this sensitivity increases as the signal to noise ratio increases.
Therefore the inference based on the assumption of normality of error can be misleading if the true error is nonnormal. These results also suggest that the MSE are sensitive to the changes in skewness and kurtosis. Lieberman b also provides the approximate Laplace distribution of. Given that the LS estimator is a ratio of quadratic forms its moments and distributions immediately follow from the results 2.
First from the result in 2.