ARIMA Processes, and the Phillips-Perron test
The ADF test requires that the order p of the AR model involved is finite, and correctly specified, i. e. the specified order should not be smaller than the actual …
Forecast error variance decomposition
In practice forecast error variance decompositions are also popular tools for interpreting VAR models. Expressing the h-step forecast error from (32.16) in terms of the orthogonalized impulse responses et = …
Nonparametric Regression
Consider the regression model V; = m(xl) + uu where i = 1,..., n, Vi is the dependent variable, x = (x;1,..., xiq) are q regressors, m(x) = E(v;|x;) is …
The Monte Carlo Test Technique
An Exact Randomized Test Procedure If there were a machine that could check 10 permutations a second, the job would run something on the order of 1,000 years. The point …
Multivariate Forecasts
The motivation for multivariate forecasting is that there is information in multiple economic time series that can be used to improve forecasts of the variable or variables of interest. Economic …
Nearly cointegrated systems
Even when a vector of time series is I(1), the size of the unit root in each of the series could be very different. For example, in terms of the …
The statistical modeling (proper) period: 1927-present
The formulation of explicit statistical models for time series began with the classic papers of Yule (1927) (Autoregressive (AR( p)) scheme): p yt = ao + X akVt-k + Ef, …
Exponential regression model
Recall that the exponential duration model depends on the parameter X, which is the constant hazard rate. We now assume an exponential distribution for each individual duration, with a rate …
Bayesian random effects model
The Bayesian fixed effects model described above might initially appeal to researchers who do not want to make distributional assumptions about the inefficiency distribution. However, as we have shown above, …
MA(q): the probabilistic reduction perspective
At this point it is important to emphasize that the above discussion relating to the convergence of certain partial sums of the MA(«) coefficients is not helpfulfrom the empirical modeling …
Characteristics of variables
The characteristics of the variables involved determine to some extent which model is a suitable representation of the data generation process (DGP). For instance, the trending properties of the variables …
Implementing the Gibbs Sampling Algorithm
Bayesian analysis of this model entails deriving the joint posterior distribution of the model's parameters and unobserved variables. Recall that the value function differences Z = {_(Zijt)j=1f2f3-i=1fN-t=1f4o} are never observed, …
Spurious Regressions with Stationary Processes
Spurious regressions in econometrics are usually associated with I(1) processes, which was explored in Phillips' well known theory and in the best known simulations. What is less appreciated is that …
Unit Root with Drift vs. Trend Stationarity
Most macroeconomic time series in (log) levels have an upwards sloping pattern. Therefore, if they are (covariance) stationary, then they are stationary around a deterministic trend. If we would conduct …
Conclusions and Extensions
Since the publication of Sims' (1980) critique of classical econometric modeling, VAR processes have become standard tools for macroeconometric analyses. A brief introduction to these models, their estimation, specification, and …
Goodness of fit measures and choices of kernel and bandwidth
The LLS estimators are easy to implement. Once a window width and kernel are chosen, Kix = K(xfx) can be computed for each value of the x = x;, j …
Monte Carlo tests in the presence of nuisance parameters
In Dufour (1995), we discuss extensions of MC tests when nuisance parameters are present. We now briefly outline the underlying methodology. In this section, n refers to the sample size …
Vector autoregressions
Vector autoregressions, which were introduced to econometrics by Sims (1980), have the form: Yt = ц t + A(L)YM + є t, (27.8) where Yt is a n x 1 …
Nonlinear error correction models
When discussing the role of the cointegrating relationship zt in (30.3) and (30.3'), we motivated the EC model as the disequilibrium mechanism that leads to the particular equilibrium. However, as …
Concluding Remarks
In the discussion of the ADF test we have assumed that the lag length p of the auxiliary regression (29.81) is fixed. It should be noted that we may choose …
The exponential model with heterogeneity
The exponential regression model can easily be extended by introducing unobservable variables. We express the individual hazard rate as: Хі = р,- exp(x!-0), (21.11) where р is a latent variable …
Summary
In this chapter, we have described a Bayesian approach to efficiency analysis using stochastic frontier models. With cross-sectional data and a log-linear frontier, a simple Gibbs sampler can be used …
ARMA Type Models: Multivariate
The above discussion of the AR( p) and MA(q) models can be extended to the case where the observable process is a vector {Zt, t Є T}, Zt : (m …
Alternative models and model representations
Given a set of K time series variables yt = (y1t,..., yKt)', the basic VAR model is of the form Vt - AiVt-i + • • • + AvVt-v + …
Experimental Design and Results
This section details the design and results of a Monte Carlo experiment that we conducted to shed light on the performance of the Gibbs sampling algorithm discussed in Section 2. …
Forecasting Economic. Time Series
James H. Stock* 1 Introduction The construction and interpretation of economic forecasts is one of the most publicly visible activities of professional economists. Over the past two decades, increased computer …
Cointegration
Juan J. Dolado, Jesus Gonzalo,and Francesc Marmol 1 Introduction A substantial part of economic theory generally deals with long-run equilibrium relationships generated by market forces and behavioral rules. Correspondingly, most …
Applications of Test Principles to Econometrics
Compared with statistics, econometrics is a relatively new discipline. Work in econometrics began in the 1920s mainly due to the initiatives of Ragnar Frisch and Jan Tinbergen. One of the …
Combined Regression
Both the parametric and nonparametric regressions, when used individually, have certain drawbacks. For example, when the a priori specified parametric regression m(x) = f (P, x) is misspecified even in …
Monte Carlo Tests: Econometric Applications
1.1 Pivotal statistics In Dufour and Kiviet (1996, 1998), Kiviet and Dufour (1997), Dufour et al. (1998), Dufour and Khalaf (1998a, 1998c), Bernard, Dufour, Khalaf, and Genest (1998), Saphores, Khalaf, …
Forecasting with leading economic indicators
Forecasting with leading economic indicators entails drawing upon a large number of time series variables that, by various means, have been ascertained to lead the variable of interest, typically taken …
Seasonal Nonstationarity and Near-Nonstationarity*
Eric Ghysels, Denise R. Osborn,and Paulo M. M. Rodrigues* 1 Introduction Over the last three decades there has been an increasing interest in modeling seasonality. Progressing from the traditional view …
More general processes
To generalize the above discussion, weakly stationary autocorrelations can be permitted in the SI(1) process. That is, (31.1) can be generalized to the seasonally integrated ARMA process: ^(L)Asyt = 0(L)et, …
Heterogeneity and negative duration dependence
The effect of unobservable covariates can be measured by comparing models with and without heterogeneity. In this section we perform such a comparison using the exponential model. For simplicity we …
The General Multivariate Parametric Problem
Consider the setting in (25.3) when { = p + v, and v ~ N(0, Q), is an available unrestricted estimator. Consider the restricted estimator p as the solution to …
