Analysis of Covariance

Dummy Variable Regression Models

(Ref: GP: Ch 9 )

INTRODUCTION

The variables that we have considered so far for the regression models are in ratio scale variables i.e. quantitative in nature (e.g. income, output, cost, price, weight which can be measured numerically). The behaviour of economics and business variables may, however, also depend on nominal scale or qualitative factors such as sex, race, colour, marital status, season so on. To represent them in the regression model some proxy must be constructed.

A dummy variable is an artificial variable constructed in such way so that it takes quantitative values unity or zero to indicate the presence or absence of a qualitative characteristic. i.e.

D = 1 if the characteristic is present,

= 0 if the characteristic is absent.

Such variables are also known as indicator variables, categorical variables, qualitative variables. We could use the dummy variables both as explanatory variables or/and dependent variable in a regression model.

Examples of Dummy Variables used as

Explanatory Variables:

1). One of the first published uses of dummy variable in economic research was to differentiate between war years and non-war years in a demand study using annual time series data.

2). In a regression model of schooling and earnings, we want to investigate whether sex or racial discrimination has any significant effect in the model.

3). In a seasonal model, whether different seasons have different effects on the dependent variable.

4) Whether any particular year government policy has any effect on inflation or employment.

Examples of Dummy Variables used as Dependent Variable:

1. Number of patents used

2. Acceptance of loan applications

3. Opinion poles or voting for a candidate

4. Consumer Choice

5. Labour force participation

USE OF DUMMY

EXPLANATORY VARIABLES

1) Analysis of Variance Models (ANOVA)

2) Analysis of Covariance Models (ANCOVA)

ANALYSIS OF VARIANCE MODEL (ANOVA)

MODELS

Regression model which contains all explanatory variables exclusively as dummy variables is known as ANOVA model. Example 9.2 in GP, p283: To find out whether average annual salary of public school teachers () differs among the three geographical regions (1) Northeast and North Central (2) South (3) West. i.e

(12.1)

where

1 if the state is in the Northeast and North

Central and 0 otherwise;

, if the state is in the south and 0 otherwise.

Note that for the three regions we are using two dummy variables and .

(Example 9.1 of GP: P280)

If a qualitative variable has m categories, we have to introduce only m-1 dummy variables to capture the effects of such categories. If this rule is not followed, we shall fall into the situation which is often known as the “Dummy Variable Trap”, the situation of perfect co-linearity or multicollinearity.

e.g. if we wanted to capture seasonal differences between electricity consumption and temperature, we should define three dummy variables due to four seasons. If we had considered and due to four quarters and intercept term in the model, then we would have exact multicollinearity problem. Since is always equal to 1 which is simply reproduced the intercept term and the least squares estimation would break down. To avoid the dummy variable trap we should define three dummy variables in this case.

The category for which no dummy variable is assigned is known as the base, benchmark, control, comparison or omitted category. The intercept represent the mean value of the benchmark category. And all comparisons are made relative to the benchmark category. The choice of the bench mark category is upto the researcher. The dummy explanatory variables estimated coefficients value in the model are the differential intercept coefficients. To get rid of dummy variable trap we could also get rid of intercept term and use all the possible dummy’s in the mode.

ANALYSIS OF COVARIANCE

MODEL (ANCOVA) MODELS

If in a regression model some explanatory variables are quantitative and some are in qualitative nature, we will obtain ANCOVA model. ANCOVA model is an extension of ANOVA model that provide a method of statistically controlling the effect of quantitative regressors or covariates or control variables. In model (12.1) including spending on public school per pupil, example of ANCOVA model is

(12.2)

Here the quantitative variable is known as covariates.

(Example 9.3 of GP: P284)

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