The probabilistic reduction perspective
The probabilistic reduction perspective contemplates the DGM of the Vector Autoregressive representation from left to right as an orthogonal decomposition: Zt = E(Zt|o(Z0-1)) + ut, t Є T, (28.28) where …
Estimation of unrestricted VARs and VECMs
Given a sample y1,..., yT and presample values y-p+1,..., y0, the K equations of the VAR (32.1) may be estimated separately by least squares (LS) without losing efficiency relative to …
Appendix A The Future Component
The future component we used was a fourth-order polynomial in the state variables. Below, in the interest of space and clarity, we will develop that polynomial only up to its …
Economic Forecasting: A Theoretical Framework
2.1 Optimal forecasts, feasible forecasts, and forecast errors Let yt denote the scalar time series variable that the forecaster wishes to forecast, let h denote the horizon of the forecast, …
Preliminaries: Unit Roots and Cointegration
2.1 Some basic concepts A well known result in time series analysis is Wold's (1938) decomposition theorem which states that a stationary time series process, after removal of any deterministic …
Models and Their Specification
Suppose the focus of the analysis is to consider the behavior of the n x 1 vector of random variables wt = (w1t, w2t,..., wnt)' observed over the period t …
Additive Regressions
In recent years several researchers have attempted to estimate m(x,) by imposing some structure upon the nature of the conditional mean m(x;). One popular solution is the generalized additive models …
Monte Carlo tests in the presence of nuisance parameters: examples from the multivariate regression model
In this section, we provide examples from Dufour and Khalaf (1998a, 1998b) pertaining to LR test criteria in the MLR (reduced form) model. The model was introduced in Section 2.3. …
Discussion and Conclusion
One of the few truly safe predictions is that economic forecasters will remain the target of jokes in public discourse. In part this arises from a lack of understanding that …
Properties of Seasonal Unit Root Processes
The case of primary interest in the context of seasonal unit roots occurs when the process yt is nonstationary and annual differencing is required to induce stationarity. This is often …
Near Seasonal Integration
As noted in Section 3.1 for the DHF test, Pr[t7s < 0] = Pr[x2(S) < S] seems to be converging to 1/2 as S increases. However, for the periodicities typically …
Truncation and censoring
Econometric data used in duration analysis are often panel data comprising a number of individuals observed over a fixed interval of time. Let us suppose that the survey concerns unemployment …
LR, W, and LM tests
We give an account of the three classical tests in the context of the general linear regression model introduced in (25.1) above. We take R to be a (p x …
Time Series and Linear Regression Models
3.3 Error autocorrelation vs. temporal dependence The classic paper of Yule (1926) had a lasting effect on econometric modeling in so far as his cautionary note that using time series …
Estimation of restricted models and structural forms
Efficient estimation of a general structural form model such as (32.7) with restrictions on the parameter matrices is more complicated. Of course, identifying restrictions are necessary for consistent estimation. In …
Appendix B Existence of Joint Posterior Distribution
Let ю denote the number of missing wage observations, and let Q = [Aq, BQ]“ £ ^++ be the domain of unobserved wages, where 0 < AQ < BQ < …
Model selection using information criteria
Because the object of point forecasting is to minimize expected loss out-ofsample, it is not desirable to minimize approximation error (bias) when this entails adding considerable parameter estimation uncertainty. Thus, …
Estimation and testing for cointegration in a single equation framework
Based upon the VECM representation, Engle and Granger (1987) suggest a two - step estimation procedure for single equation dynamic modeling which has become very popular in applied research. Assuming …
Semiparametric models
A partial solution to the dimensionality problem was also explored in Speckman (1988) and Robinson (1988). They considered the case where xi = (xi1, xi2) and m(x) = m(xi1, x12) …
Non-identified nuisance parameters
The example we discuss here is the problem of testing for the significance of jumps in the context of a jump-diffusion model. For econometric applications and references, see Saphores et …
Time Series and Dynamic Models
Aris Spanos 1 Introduction This chapter discusses certain dynamic statistical models of interest in modeling time series data. Particular emphasis is placed on the problem of statistical adequacy: the postulated …
Asymptotic properties
Consider the DGP of the seasonal random walk with initial values y-§+s =... = y0 = 0. Using the notation of (31.6), the following § independent partial sum processes (PSPs) …
Specifying the cointegrating rank
In practice, the cointegrating rank r is also usually unknown. It is commonly determined by a sequential testing procedure based on likelihood ratio (LR) type tests. Because for a given …
Proportional hazard model
In this model the conditional hazard functions are assumed homothetic and the parametric term exp(x,0) is introduced as the coefficient of proportionality: X( Уі 1 xi; ^ f0) = exp(x0)X …
Asymtotically normal estimators
The assumption of normality can be relaxed for the situations that allow asymptotically normal estimators for p. This is because the inference theory developed for problems (25.3) or (25.8) is …
Dynamic linear regression models
The dynamic linear regression model (28.35) discussed above, constitutes a reduction of the VAR(1) model (28.31), given that we can decompose D(Z t|Z t-1; ф) further, based on the separation …
Model Specification and Model Checking
3.1 Choosing the model order Unrestricted VAR models usually involve a substantial number of parameters which in turn results in rather imprecise estimators. Therefore, it is desirable to impose restrictions …
Monte Carlo Test. Methods in. Econometrics
Jean-Marie Dufour and Lynda Khalaf [14] 1 Introduction During the last 20 years, computer-based simulation methods have revolutionized the way we approach statistical analysis. This has been made possible by …
Prediction intervals
In some cases, the object of forecasting is not to produce a point forecast but rather to produce a range within which yt+h has a prespecified probability of falling. Even …
System-Based Approaches to Cointegration
Whereas in the previous section we confined the analysis to the case where there is at most a single cointegrating vector in a bivariate system, this setup is usually quite …
Hypothesis Testing
An obvious question is how to carry out various diagnostic tests done in the parametric econometrics within the nonparametric and semiparametric models. Several papers have appeared in the recent literature …
Bayesian Analysis. of Stochastic. Frontier Models
Gary Koop and Mark F. J. Steel 1 Introduction Stochastic frontier models are commonly used in the empirical study of firm1 efficiency and productivity. The seminal papers in the field …
Time series: a brief historical introduction
Time series data have been used since the dawn of empirical analysis in the mid-seventeenth century. In the "Bills of Mortality" John Graunt compared data on births and deaths over …
Deterministic seasonality
A common practice is to attempt the removal of seasonal patterns via seasonal dummy variables (see, for example, Barsky and Miron, 1989; Beaulieu and Miron, 1991; Osborn, 1990). The interpretation …
Duration Time Series
In this section we focus our attention on duration time series, i. e. sequences of random durations, indexed by their successive numbers in the sequence and possibly featuring temporal dependence. …
