Residual Interpretation of Multiple Regression Estimates
Although we did not derive an explicit solution for the OLS estimators of the P’s, we know that they are the solutions to (4.2) or (4.3). Let us focus on …
Critiques of Econometrics
Econometrics has its critics. Interestingly, John Maynard Keynes (1940, p. 156) had the following to say about Jan Tinbergen’s (1939) pioneering work: No one could be more frank, more painstaking, …
Infinite Distributed Lag
So far we have been dealing with a finite number of lags imposed on Xt. Some lags may be infinite. For example, the investment in building highways and roads several …
Statistical Properties of Least Squares
(i) Unbiasedness Given assumptions 1-4, it is easy to show that вOLS is unbiased for в. In fact, using equation (3.4) one can write Pols = EEi xW EEi xj …
Recursive Residuals
In Section 8.1, we showed that the least squares residuals are heteroskedastic with non-zero covariances, even when the true disturbances have a scalar covariance matrix. This section studies recursive residuals …
Overspecification and Underspecification of the Regression Equation
So far we have assumed that the true linear regression relationship is always correctly specified. This is likely to be violated in practice. In order to keep things simple, we …
Looking Ahead
Econometrics have experienced phenomenal growth in the past 50 years. There are six volumes of the Handbook of Econometrics, most of it dealing with post 1960’s research. A lot of …
Estimation and Testing of Dynamic Models with Serial Correlation
Both the AEM and the PAM give equations resembling the autoregressive form of the infinite distributed lag. In all cases, we end up with a lagged dependent variable and an …
Estimation of a2
The variance of the regression disturbances o2 is unknown and has to be estimated. In fact, both the variance of вOLS and that of aOLS depend upon о2, see (3.6) …
Applications of Recursive Residuals
Recursive residuals have been used in several important applications: (1) Harvey (1976) used these recursive residuals to give an alternative proof of the fact that Chow’s post-sample predictive test has …
R-Squared Versus R-Bar-Squared
Since OLS minimizes the residual sums of squares, adding one or more variables to the regression cannot increase this residual sums of squares. After all, we are minimizing over a …
Basic Statistical Concepts
2.1 Introduction One chapter cannot possibly review what one learned in one or two pre-requisite courses in statistics. This is an econometrics book, and it is imperative that the student …
A Lagged Dependent Variable Model with AR(1) Disturbances
A model with a lagged dependent variable and an autoregressive error term is estimated using instrumental variables (IV). This method will be studied extensively in Chapter 11. In short, the …
Maximum Likelihood Estimation
Assumption 5: The ui’ s are independent and identically distributed N(0,a2). This assumption allows us to derive distributions of estimators and other test statistics. In fact using (3.5) one can …
Specification Tests
Specification tests are an important part of model specification in econometrics. In this section, we only study a few of these diagnostic tests. For an excellent summary on this topic, …
Testing Linear Restrictions
In the simple linear regression chapter, we proved that the OLS estimates are BLUE provided assumptions 1 to 4 were satisfied. Then we imposed normality on the disturbances, assumption 5, …
Methods of Estimation
Consider a Normal distribution with mean ц and variance a2. This is the important “Gaussian” distribution which is symmetric and bell-shaped and completely determined by its measure of centrality, its …
Autoregressive Distributed Lag
So far, section 6.1 considered finite distributed lags on the explanatory variables, whereas section 6.2 considered an autoregressive relation including the first lag of the dependent variable and current values …
A Measure of Fit
We have obtained the least squares estimates of a, в and a2 and found their distributions under normality of the disturbances. We have also learned how to test hypotheses regarding …
Dummy Variables
Many explanatory variables are qualitative in nature. For example, the head of a household could be male or female, white or non-white, employed or unemployed. In this case, one codes …
Properties of Estimators
(i) Unbiasedness Д is said to be unbiased for ц if and only if EQX) = ц For Д = X, we have E(X) = £™=1 E(Xi)/n = ц and …
The General Linear Model: The Basics
7.1 Introduction Consider the following regression equation y = Хв + u (7.1) where ' Yi " ' Xu X12 • • Xik ■ ' в i " u1 y …
Prediction
Let us now predict Y0 given X0. Usually this is done for a time series regression, where the researcher is interested in predicting the future, say one period ahead. This …
Violations of the Classical Assumptions
5.1 Introduction In this chapter, we relax the assumptions made in Chapter 3 one by one and study the effect of that on the OLS estimator. In case the OLS …
Hypothesis Testing
The best way to proceed is with an example. Example 1: The Economics Departments instituted a new program to teach micro-principles. We would like to test the null hypothesis that …
Partitioned Regression and the Frisch-Waugh-Lovell Theorem
In Chapter 4, we studied a useful property of least squares which allows us to interpret multiple regression coefficients as simple regression coefficients. This was called the residualing interpretation of …
Residual Analysis
A plot of the residuals of the regression is very important. The residuals are consistent estimates of the true disturbances. But unlike the ui’s, these ei’s are not independent. In …
Stochastic Explanatory Variables
Sections 5.5 and 5.6 will study violations of assumptions 2 and 3 in detail. This section deals with violations of assumption 4 and its effect on the properties of the …
Likelihood Ratio, Wald and Lagrange Multiplier Tests
Before we go into the derivations of these three tests we start by giving an intuitive graphical explanation that will hopefully emphasize the differences among these tests. This intuitive explanation …
Confidence Intervals and Test of Hypotheses
We start by constructing a confidence interval for any linear combination of в, say c' в. We know that cCвOLS ~ N(с'в,&2с'(X'X)-1c) and it is a scalar. Hence, Zobs = …
Numerical Example
Table 3.1 gives the annual consumption of 10 households each selected randomly from a group of households with a fixed personal disposable income. Both income and consumption are measured in …
Normality of the Disturbances
If the disturbance are not normal, OLS is still BLUE provided assumptions 1-4 still hold. Normality made the OLS estimators minimum variance unbiased MVU and these OLS estimators turn out …
Confidence Intervals
Estimation methods considered in section 2.2 give us a point estimate of a parameter, say л, and that is the best bet, given the data and the estimation method, of …
Joint Confidence Intervals and Test of Hypotheses
We have learned how to test any single hypothesis involving any linear combination of the в’s. But what if we are interested in testing two or three or more hypotheses …
