ECONOMETRICS Lecturer Dr. Veronika Alhanaqtah

Topic 3. Topic 3. Nonlinear regressions Selection of functional forms of models and problems of specification Logarithmic (log-linear) models Semi-log models Dummy variables in regression analysis Selection of a functional form for a model. ◦ Features of a “good” model ◦ Types of specification problems ◦ Detection of specification problems and its elimination. F-test  Testing hypothesis on some linear restrictions simultaneously  Testing hypothesis on insignificance of a regression as a whole Ramsey-test (RESET) Quality criteria of a regression model

Linear and nonlinear models application

1. Logarithmic (log-linear) models

2. Semi-log models

3. Dummy variables in regression analysis Not only numeric (quantitative) variables, but also categorical (factor, qualitative) variables. For example, demand for a specific good can be determined by its price, price on substitution-good, consumer’s income, etc. But demand for the good can also be determined by consumer’s tastes and expectations, national and religion peculiarities, etc. Problem: how to determine the influence of categorical variables on a dependent variable? Plug into models a dummy variable which represents two opposite qualities of a categorical variable.

3. Dummy variables in regression analysis

ANOVA-models are regression models, which contain only categorical variables. (models of dispersion analysis) ANCOVA-models are regression models, which contain both numeric and categorical variables. (models of covariance analysis)

3. Dummy variables in regression analysis

Dummy variables in seasonal analysis Many economic variables ate directly influenced by seasonal fluctuations. For example, demand for tourist trips, cold drinks and ice-cream are higher in summer than in winter; demand for warm clothes is higher in winter. How do we include dummy variables in this case?

3. Dummy variables in regression analysis

observationseasonspring i summer i autumn i 1winter000 2spring100 3summer010 4autumn001 5winter000 ……………

3. Dummy variables in regression analysis

Diversity and complexity of economic processes predetermine a large variety of models, used in econometric analysis. In case of a simple linear regression, model selection is usually based upon the layout of points plotted on a correlation field. It often happens, that layout of points approximately corresponds to several functions, and our task is to find the most adequate. For example, curvilinear relationships could be approximated by polynomial, power, exponential, logarithmic functions. For multiple regression plotting of statistical data is impossible.

3. Dummy variables in regression analysis Remember: an ideal model does not exist. Questions: What are the features of a “good”, high-quality, model? What specification problems we might face to and what are the consequences of these problems? How to find out a specification problem? How to eliminate a specification problem and to move on to the better (more qualitative) model?

4.1. Features of a “good” model Simplicity. Out of 2 models, reflecting the reality approximately similarly, we’d rather chose a model with fewer number of variables. Uniqueness. For any set of statistical data, estimates of coefficients must be unique (with a single meaning). The right match. Equation (model) is admitted to be better, if it can explain more variance of a dependent variable, in comparison with other models. Choose the regression model with a higher R-squared-adjusted. Reconciliation with theory. For example, if in a demand function a coefficient at price is appeared to be positive, the model is not admitted to be “good”, even though it has high R 2 (like 0.7). A model must be based on theoretical grounding. Prognostic qualities. Model is of high quality if its predictions are acknowledged by real data.

4.2. Types of specification problems Good specification of a regression means that a regression, on the whole, reflects a relationship between economic parameters adequately. Incorrect selection of a functional form of a model or incorrect selection of a set of independent variables (regressors) is called problems of specification.

4.2. Types of specification problems

(3) Selection of incorrect functional form of a model. This mistake is very serious. Prediction qualities of a model are very low.

4.3. Detection of specification problems and its elimination. F-test One insignificant variable: revealed by low t-statistic. Several insignificant variables: then we have to construct a regression model without these variables. Using F-statistic, compare R 2 for unrestricted (initial) model and restricted model (without insignificant variables).

Testing hypothesis on some linear restrictions simultaneously Example 3 Researcher estimated a relationship between price of an accommodation in Moscow (Russia), dependent on some factors. He included in the initial model (UR-unrestricted) the following factors: total space, living space, kitchen space, whether an apartment is in a brick or non-brick building (“1” for brick, “0” for non-brick), distance to the metro station, whether you can reach metro walking or driving (“1” for walking). He also considered another model (R-restricted), where he omitted some variables. Number of observations is Which model is better?

Testing hypothesis on some linear restrictions simultaneously

A little bit mathematical theory. Sum of squares

Testing hypothesis on insignificance of a regression as a whole

Moral for the Question : If it is supposed in the theory that y is dependent on z, it is recommended to include z in the model, even if z is insignificant. If variables are significant, it is better to include them in the model, even if in the theory it is not supposed any dependence on them.

Other tests for identification of specification problems Ramsey-test (RESET-Regression specification error test) The Likelihood Ration test The Wald test The Lagrange multiplier test The Hausman test Box-Cox transformation.

5. Ramsey-test (RESET)

6. Quality criteria of a regression model

Homework Visit instructor’s web-page on Econometrics. Do the Homework test.