Econometrics 3rd yr PDF

Introduction to Econometrics compiled by: PaulosT Chapter One Definition, Scope & Types of Econometrics  What is econometrics? 1.1 Introduction Definition: Econometrics deals with the measurement of economic relations...

Introduction to Econometrics compiled by: PaulosT Chapter One Definition, Scope & Types of Econometrics  What is econometrics? 1.1 Introduction Definition: Econometrics deals with the measurement of economic relationships. Econometrics is a combination of economic theory, mathematical economics and statistics, but it is completely distinct from each one of these three branches of science. The relationships and differences among these sciences are pointed out below. A. Economic theory makes statements or hypotheses that are mostly qualitative in nature Ex. Microeconomic theory states that, other things remaining the same, a reduction in the price of a commodity is expected to increase the quantity demanded of that commodity. But the theory itself does not provide any numerical measure of the relationship between the two: that is it does not tell by how much the quantity will go up or down as a result of a certain change in the price of the commodity. It is the job of econometrician to provide such numerical statements. B. Mathematical economics: the main concern of Mathematical economics is to express economic theory in mathematical form (equations) without regard to measurability or empirical verification of the theory. Both economic theory and mathematical economics state the same relationships. Economic theory uses verbal exposition but mathematical economics employs mathematical symbolism. Neither of them allows for random elements which might affect the relationship and make it stochastic. Furthermore, they do not provide numerical values for the coefficients of the relationships. Although econometrics presupposes the expression of economic relationships in mathematical form, like mathematical economics it does not assume that economic relationships are exact (deterministic). It assumes that relationships are not exact Econometric methods are designed to take in to account random disturbances which create deviations from the exact behavioral patterns suggested by economic theory and mathematical economics. Econometrics provides numerical values of the coefficients of economic phenomena. C. Economic Statistics is mainly concerned with collecting, processing, and presenting economic data in the form of charts and tables. It is mainly a descriptive aspect of economics. It does not provide explanations of the development of the various variables and it does not provide measurement of the parameters of economic relationships. The econometrician often needs special methods since the data are not generated as the result of a controlled experiment. This creates special problems not normally dealt with in By: PaulosT. 1  Introduction to Econometrics compiled by: PaulosT mathematical statistics. Moreover, such data are likely to contain errors of measurement, and the econometrician may be called up on to develop special methods of analysis to deal with such errors of measurement. Note: To conclude econometrics is an amalgam of economic theory, mathematical economics, economic statistics, and mathematical statistics. Yet, it is a subject that deserves to be studied in its own right for the above mentioned reasons. 1.2 Goals of Econometrics Three main goals of econometrics 1. Analysis: - Testing Economic Theory Economists formulated the basic principles of the functioning of the economic system using verbal exposition and applying a deductive procedure. Economic theories thus developed in an abstract level were not tested against economic reality. Econometrics aims primarily at the verification of economic theories. 2. Policy-Making In many cases we apply the various econometric techniques in order to obtain reliable estimates of the individual coefficients of the economic relationships from which we may evaluate elasticities or other parameters of economic theory (multipliers, technical coefficients of production, marginal costs, marginal revenues, etc.) The knowledge of the numerical value of these coefficients is very important for the decisions of firms as well as for the formulation of the economic policy of the government. It helps to compare the effects of alternative policy decisions. 3. Forecasting In formulating policy decisions it is essential to be able to forecast the value of the economic magnitudes. Such forecasts will enable the policy-maker to judge whether it is necessary to take any measures in order to influence the relevant economic variables. 1.3 TYPES OF ECONOMETRICS Econometrics may be divided in to two broad categories 1. Theoretical Econometrics By: PaulosT. 2  Introduction to Econometrics compiled by: PaulosT 2. Applied Econometrics 1. Theoretical Econometrics is concerned with the development of appropriate methods for measuring economic relationships specified by econometric models. In this aspect, econometrics tends heavily on mathematical statistics. For example, one of the tools that are used extensively is the method of least squares. It is the concern of theoretical econometrics to spell out the assumptions of this method, its properties, and what happens to these properties when one or more of the assumptions of the method are not fulfilled. 2. In Applied Econometrics we use the tools of theoretical econometrics to study some special field(s) of economics, such as the production function, consumption function, investment function, demand and supply functions, etc. Applied econometrics includes the applications of econometric methods to specific branches of economic theory. It involves the application of the tools of theoretical econometrics for the analysis of economic phenomena and forecasting economic behavior. 1.4 Methodology of Econometrics In any econometric research we may distinguish four stages: A. Specification of the model The first, and the most important, step the econometrician has to take in attempting the study of any relationship between variables is to express this relationship in mathematical form, that is to specify the model, with which the economic phenomenon will be explored empirically. This is called the specification of the model or formulation of the maintained hypothesis. It involves the determination of: i. The dependent and explanatory variables which will be included in the model. The econometrician should be able to make a list of the variables that might influence the dependent variable.  General economic theories,  Previous studies in any particular field and  Information about individual condition in a particular case, and the actual behavior of the economic agents may indicate the general factors that affect the dependent variable.  The a priori theoretical expectations about the sign and the size of the parameters of the function. These a priori definitions will be the theoretical criteria on the basis of which the results of the estimation of the model will be evaluated  Economic theory  Other applied research  Information about possible special features of the phenomena being studied will contain suggestions about the sign and size of the parameters. By: PaulosT. 3  Introduction to Econometrics compiled by: PaulosT Example: Consider the following simple consumption function: C = 0 + 1Y+ U Where: C = Consumption function Y = level of income In this function the coefficient 1 is the marginal propensity to consume (MPC) and should be positive with a value less than unity (0 tc , reject H0 and accept H1. The conclusion is ˆ is statistically significant.  If t*< tc , accept H0 and reject H1. The conclusion is ˆ is statistically insignificant. Numerical Example: Suppose that from a sample size n=20 we estimate the following consumption function: C  100  0.70  e (75.5) (0.21) The values in the brackets are standard errors. We want to test the null hypothesis: H 0 :  i  0 against the alternative H1 :  i  0 using the t-test at 5% level of significance. a. the t-value for the test statistic is: ˆ  0 ˆ 0.70 t*   =  3.3 SE ( ˆ ) SE ( ˆ ) 0.21 b. Since the alternative hypothesis (H1) is stated by inequality sign (  ) ,it is a two tail test, hence we divide  2  0.05 2  0.025 to obtain the critical value of ‘t’ at  2 =0.025 and 18 degree of freedom (df) i.e. (n-2=20-2). From the t-table ‘tc’ at 0.025 level of significance and 18 df is 2.10. By: PaulosT. 47  Introduction to Econometrics compiled by: PaulosT c. Since t*=3.3 and tc=2.1, t*>tc. It implies that ˆ is statistically significant. i ii) Confidence interval Rejection of the null hypothesis doesn’t mean that our estimate ˆ and ˆ is the correct estimate of the true population parameter  and . It simply means that our estimate comes from a sample drawn from a population whose parameter  is different from zero. In order to define how close the estimate to the true parameter, we must construct confidence interval for the true parameter, in other words we must establish limiting values around the estimate with in which the true parameter is expected to lie within a certain “degree of confidence”. In this respect we say that with a given probability the population parameter will be with in the defined confidence interval (confidence limits). We choose a probability in advance and refer to it as confidence level (interval coefficient). It is customarily in econometrics to choose the 95% confidence level. This means that in repeated sampling the confidence limits, computed from the sample, would include the true population parameter in 95% of the cases. In the other 5% of the cases the population parameter will fall outside the confidence interval. In a two-tail test at  level of significance, the probability of obtaining the specific t-value either –tc or tc is  2 ˆ   at n-2 degree of freedom. The probability of obtaining any value of t which is equal to at n-2 degree of SE ( ˆ ) freedom is 1   2   2  i.e. 1  . i.e. Pr t c  t*  t c   1   …………………………………………(2.57) ˆ   but t*  …………………………………………………….(2.58) SE ( ˆ ) Substitute (2.58) in (2.57) we obtain the following expression.  ˆ    Pr  t c   t c   1   ………………………………………..(2.59)  SE( ˆ )    Pr  SE(ˆ )t c  ˆ    SE(ˆ )t c  1        by multiplying SE(ˆ ) Pr ˆ  SE(ˆ )t    ˆ  SE(ˆ )t  1        by subtracting ˆ c c Pr ˆ  SE(ˆ )    ˆ  SE(ˆ )t  1        by multiplying by  1 c Prˆ  SE(ˆ )t    ˆ  SE(ˆ )t  1        int erchanging c c By: PaulosT. 48  Introduction to Econometrics compiled by: PaulosT The limit within which the true  lies at (1   )% degree of confidence is: [ˆ  SE(ˆ )t c , ˆ  SE(ˆ )t c ] ; where tc is the critical value of t at  2 confidence interval and n-2 degree of freedom. The test procedure is outlined as follows. H0 :   0 H1 :   0 Decision rule: If the hypothesized value of  in the null hypothesis is within the confidence interval, accept H0 and reject H1. The implication is that ˆ is statistically insignificant; while if the hypothesized value of  in the null hypothesis is outside the limit, reject H0 and accept H1. This indicates ˆ is statistically significant. Numerical Example: Suppose we have estimated the following regression line from a sample of 20 observations. Y  128.5  2.88 X  e (38.2) (0.85) The values in the bracket are standard errors. a. Construct 95% confidence interval for the slope of parameter b. Test the significance of the slope parameter using constructed confidence interval. Solution: a. The limit within which the true  lies at 95% confidence interval is: ˆ  SE(ˆ )t c ˆ  2.88 SE(ˆ )  0.85 t c at 0.025 level of significance and 18 degree of freedom is 2.10.  ˆ  SE(ˆ )t c  2.88  2.10(0.85)  2.88  1.79. The confidence interval is: (1.09, 4.67) b. The value of  in the null hypothesis is zero which implies it is out side the confidence interval. Hence  is statistically significant. 2.2.3 Reporting the Results of Regression Analysis By: PaulosT. 49  Introduction to Econometrics compiled by: PaulosT The results of the regression analysis derived are reported in conventional formats. It is not sufficient merely to report the estimates of  ’s. In practice we report regression coefficients together with their standard errors and the value of R2. It has become customary to present the estimated equations with standard errors placed in parenthesis below the estimated parameter values. Sometimes, the estimated coefficients, the corresponding standard errors, the p-values, and some other indicators are presented in tabular form. These results are supplemented by R2 on ( to the right side of the regression equation). Y  128.5  2.88 X Example: , R2 = 0.93. The numbers in the parenthesis below the parameter (38.2) (0.85) estimates are the standard errors. Some econometricians report the t-values of the estimated coefficients in place of the standard errors. By: PaulosT. 50