Lecture 1 - econometric problems PDF

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Lecture 1: Recap: Econometrics Is the interaction of economic theory, observed data and statistical methods, i.e. quantify the relationship on the basis of observed data. Historically, econometrics started with macro data (measure the relationship between aggregate economic data). Later on (around 1970s) microeconometric models describing individuals, households and firms decision started to be developed. The foundations of the relationships are in mathematical terms which lead to econometric or statistical models. Normally three types of models are studied which are cross-sectional models (like household decisions on consumption in a week), time series models (studying the relationship of variables overtime) and panel data models (observing the same units over a period of time). Knowing economic theory is different from specifying a particular model, usually model specification is justified by economic theory and the nature of data. Examples of common econometric models are OLS, limited dependent variable models and vector autoregressive models. The simplest econometric models is the ordinary least square model (OLS). This model minimizes the sum of squared errors (deviation between actual values and estimated values of the dependent variable). That is Yi = a0 + a1x1i + ……… + anxni + ε

i

…… ( actual relationship)

Then a0 + a1x1i + ……… + anxni is referred as estimated y, denoted as Therefore yi - ^y i= ε

i

^y

is referred as residuals or errors.

The general ideal of OLS is to minimize squared residuals or errors. The general procedure is all algebraic formulation. No economic theory or statistical theory that enters into picture.

For economic relationship to be in the picture, several assumptions has to be satisfied. These assumptions technically are known as Gauss-Markow Assumptions. When these assumptions are satisfied we have The Linear Regression Model (LRM). Some of these assumptions includes: 

Linearity in parameters



No multicollinearity; that is r(x)=k where k=number of parameters



Exogeneity of regressors (regressors are exogeneous). This ensure the accuracy of the OLS estimators particularly unbiasedness



Spherical disturbances, variances of error terms is spherical which rules out heteroscedasticity and autocorrelation of the error terms. This ensure the precision of the OLS estimators, that is with small variance



The error terms are normally distributed which justify the use of t and F statistics in inferences

There are many assumptions that has to be satisfied for OLS to produce the best estimates of the relationship between economic variables, that is OLS to be blue. When these estimates are violated we get what is referred as econometrics problems. In this course we will only focus on three econometric problems which are heteroscedasticity, autocorrelation and multicollinearity. Heteroscedasticity and autocorrelation These are econometric problems that arises when the assumption of spherical disturbances is violated, that is the variance of error terms is not constant and correlation between error terms of observation is not equal to zero. There are three main problems which might arise when this assumption is violated. 

Pure heteroscedasticity, variance of error terms differ across observations



Pure serial correlation, variance of errors terms is constant across observations but correlation of error terms across observations is not equal to zero.



Both problems, variance of error terms is not constant and correlation is not equal to zero across observations.

Heteroscedasticity is very common in cross-sectional data while autocorrelation is very common in time series data although both problems can happen in both types of data. Why do we have to worry about these problems? The two problems have the same consequences, the OLS estimators remain accurate (that is unbiased). The only worry is on variance of OLS estimators. Due to heteroscedasticity and/or autocorrelation, OLS estimators will have large variances and larger standard errors and thus become inefficient. They affect the distribution of OLS estimators. The OLS estimators will no longer be efficient. Due to such problem, the standard errors are going to be invalid, the t and F statistics are going to be invalid and hence wrong or invalid inferences. Why? Because we use a wrong formula in calculating the variance of OLS estimators. How do we detect the problem? There are two methods, informal and formal methods.  Informal methods involves plotting residuals (which are used as the proxy of error terms) 

For autocorrelation, we plot residuals against time, there should be no specific trend( either down ward or upward), otherwise there is autocorrelation



Heteroscedasticity, we plot residuals against estimated values or observations, the distance between mean value of the residuals and residuals should be constant.

 Formal methods o Autocorrelation 

Durbin Watson test, most applied in order one, AR(1)



Breush-Godfrey test, applied at any order

o Heteroscedasticity 

Breush-Pagan test. This assumes error terms varies with explanatory variables(residuals can be used as the proxy). Estimate the model with residuals as an independent variable and use F test to test the overall significance of the model.



White test(Halbert white). Include the squares and the cross-product and extend the F-test and use the same above framework. Sometimes dependent variable and its square is used due to the complexity in parameters

How do we solve the problem? There are two methods used, generalized least square and weighted least square. Inhere we will focus on generalized least square. The general idea behind GLS is to find a multiple factor to transform an error term from the one with Heteroscedasticity and/or autocorrelation to the one without such a problem...