Log Dummy Regression Interpretation

[Reda]

Iam wondering how I can interpret the estimated coefficient for variable B. I thought I could run a simple regression analysis, but I would also like to get an estimated effect of variable A while B is zero (which is what the coefficient for variable A represents, if I am understanding correctly).

The way you are interpreting the coefficients is not quite right. The general interpretation of the coefficient on a dummy variable in a multiple regression is "the expected (or average) difference in the dependent variable between those with $1$ and those with $0$ values of that dummy variable, holding other independent variables constant.

Variable A can be present (i.e., 1) only when Variable B is present (1). I am wondering how I can interpret the estimated coefficient for variable B, because the coefficient for B represent the presence of B while A is 0, which logically does not make sense.

Thus, when we have an intercept in the regression model and we want to avoid perfect multicollinearity, we create only one dummy to encode a categorical variable that has two categories.

I want to look at bivariate relationships among a binary outcome and multiple predictor variables before conducting multiple regression. EDIT: [The data are multiply imputed] and the categorical predictors have been dummy coded. Therefore, for each nominal categorical variable in the original data, there are k-1 dummy variables, with the omitted category serving as the reference group.

It seemed to me that there was no meaningful "bivariate" relationship between the outcome and a single [dummy] variable from a collection of related variables (this would change the comparison category to "everything else," which could not be the case in a multiple regression). For this reason, I did not use bivariate correlations. Instead, I did a series of "bivariate" logistic regressions including in a single regression all k-1 dummies of a variable. So, like: outcome = race_black race_other (race_white omitted). In any case, the reference category cannot be examined directly since it was omitted in the multiple imputation process and has missing values.

After laboriously compiling a table of the results of these "bivariate" regressions without including the intercept, it occurred to me that (MAYBE?) the intercept is the "dummy" for the omitted category. If it is significant, is it in fact indicating that the omitted group is different from the mean on the outcome? If that's true, should I include the variable in my omnibus regression even if none of the explicit category variables are significant predictors?

I think you are making this hard on yourself. Make sure race is a factor variable so that the software provides the overall $\chi^2$ of association with $k-1$ d.f. for $k$ categories. Coding doesn't affect the value of $\chi^2$. Don't use a stepwise process for making inference about the importance of race. Use the overall "chunk" test as described above, which has a built-in perfect multiplicity adjustment besides being invariant to coding. In R this would look like (for a binary or ordinal logistic model predicting $Y$):

@FrankHarrell explained how to test whether race has an effect. You may also be interested in the size of that overall effect, especially since often it would be very implausible that the null hypothesis is true, that is, that a variable like race has exactly no effect whatsoever. So if we reject the null hypothesis we do not add much to our knowledge; we already knew that the null hypothesis was false before we analysed the data. If we fail to reject the null hypothesis, then that just meant that our dataset was not large enough to detect the effect that really exists and that to is not an overly informative conclussion. Instead a single summary measure of how large the effect of race could add a lot more, and this is what a sheaf coefficient was designed to do. In fact race/ethnicity is one of the examples used in the original article.

I am not sure about my interpretation of the t-ratios in dummy regression models for event studies. I have the results for two different groups of models examining the impact of news on stock returns and I want to compare them.

I am not quit understand what do you mean by absolute abnormal returns. I can understand you want to test the asymmetric effects of good news and bad news. There is a similar test named sign bias test by Engle, exactly, which is applied in the GARCH model, you can check it.

Technically, dummy variables are dichotomous, quantitative variables. Their range of values is small; they can take on only two quantitative values. As a practical matter, regression results are easiest to interpret when dummy variables are limited to two specific values, 1 or 0. Typically, 1 represents the presence of a qualitative attribute, and 0 represents the absence.

The number of dummy variables required to represent a particular categorical variable depends on the number of values that the categorical variable can assume. To represent a categorical variable that can assume k different values, a researcher would need to define k - 1 dummy variables.

For example, suppose we are interested in political affiliation, a categorical variable that might assume three values - Republican, Democrat, or Independent. We could represent political affiliation with two dummy variables:

In this example, notice that we don't have to create a dummy variable to represent the "Independent" category of political affiliation. If X1 equals zero and X2 equals zero, we know the voter is neither Republican nor Democrat. Therefore, voter must be Independent.

When defining dummy variables, a common mistake is to define too many variables. If a categorical variable can take on k values, it is tempting to define k dummy variables. Resist this urge. Remember, you only need k - 1 dummy variables.

A kth dummy variable is redundant; it carries no new information. And it creates a severe multicollinearity problem for the analysis. Using k dummy variables when only k - 1 dummy variables are required is known as the dummy variable trap. Avoid this trap!

The value of the categorical variable that is not represented explicitly by a dummy variable is called the reference group. In this example, the reference group consists of Independent voters.

In analysis, each dummy variable is compared with the reference group. In this example, a positive regression coefficient means that income is higher for the dummy variable political affiliation than for the reference group; a negative regression coefficient means that income is lower. If the regression coefficient is statistically significant, the income discrepancy with the reference group is also statistically significant.

In this section, we work through a simple example to illustrate the use of dummy variables in regression analysis. The example begins with two independent variables - one quantitative and one categorical. Notice that once the categorical variable is expressed in dummy form, the analysis proceeds in routine fashion. The dummy variable is treated just like any other quantitative variable.

The first thing we need to do is to express gender as one or more dummy variables. How many dummy variables will we need to fully capture all of the information inherent in the categorical variable Gender? To answer that question, we look at the number of values (k) Gender can assume. We will need k - 1 dummy variables to represent Gender. Since Gender can assume two values (male or female), we will only need one dummy variable to represent Gender.

Note that X1 identifies male students explicitly. Non-male students are the reference group. This was a arbitrary choice. The analysis works just as well if you use X1 to identify female students and make non-female students the reference group.

At this point, we conduct a routine regression analysis. No special tweaks are required to handle the dummy variable. So, we begin by specifying our regression equation. For this problem, the equation is:

Values for IQ and X1 are known inputs from the data table. The only unknowns on the right side of the equation are the regression coefficients, which we will estimate through least-squares regression.

The first task in our analysis is to assign values to coefficients in our regression equation. Excel does all the hard work behind the scenes, and displays the result in a regression coefficients table:

The coefficient of muliple determination is 0.810. For our sample problem, this means 81% of test score variation can be explained by IQ and by gender. Translation: Our equation fits the data pretty well.

Before we conduct those tests, however, we need to assess multicollinearity between independent variables. If multicollinearity is high, significance tests on regression coefficient can be misleading. But if multicollinearity is low, the same tests can be informative.

To measure multicollinearity for this problem, we can try to predict IQ based on Gender. That is, we regress IQ against Gender. The resulting coefficient of multiple determination (R2k) is an indicator of multicollinearity. When R2k is greater than 0.75, multicollinearity is a problem.

With multiple regression, there is more than one independent variable; so it is natural to ask whether a particular independent variable contributes significantly to the regression after effects of other variables are taken into account. The answer to this question can be found in the regression coefficients table:

The regression coefficients table shows the following information for each coefficient: its value, its standard error, a t-statistic, and the significance of the t-statistic. In this example, the t-statistics for IQ and gender are both statistically significant at the 0.05 level. This means that IQ predicts test score beyond chance levels, even after the effect of gender is taken into account. And gender predicts test score beyond chance levels, even after the effect of IQ is taken into account.

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