Use the data in MEAP9 3 to answer this question. (i)...

Use the data in MEAP9 3 to answer this question. (i) Estimate the model $$m a t h l 0=\beta_{0}+\beta_{1} \log (\text {expend})+\beta_{2} \text { lnchprg }+u$$ and report the results in the usual form, including the sample size and $R$ -squared. Are the signs of the slope coefficients what you expected? Explain. (ii) What do you make of the intercept you estimated in part (i)? In particular, does it make sense to set the two explanatory variables to zero? [Hint: Recall that log $(1)=0 . ]$ (iii) Now run the simple regression of $m a t h 10$ on log(expend), and compare the slope coefficient with the estimate obtained in part (i). Is the estimated spending effect now larger or smaller than in part (i)? (iv) Find the correlation between lexpend $=$ log $($expend$)$ and lnchprg. Does its sign make sense to you? (v) Use part (iv) to explain your findings in part (iii).

Use the data in MEAP9 3 to answer this question.
(i) Estimate the model
$$m a t h l 0=\beta_{0}+\beta_{1} \log (\text {expend})+\beta_{2} \text { lnchprg }+u$$
and report the results in the usual form, including the sample size and $R$ -squared. Are the signs
of the slope coefficients what you expected? Explain.
(ii) What do you make of the intercept you estimated in part (i)? In particular, does it make sense to
set the two explanatory variables to zero? [Hint: Recall that log $(1)=0 . ]$
(iii) Now run the simple regression of $m a t h 10$ on log(expend), and compare the slope coefficient with the estimate obtained in part (i). Is the estimated spending effect now larger or smaller than in part (i)?
(iv) Find the correlation between lexpend $=$ log $($expend$)$ and lnchprg. Does its sign make sense to you?
(v) Use part (iv) to explain your findings in part (iii).

Key Concepts

Multiple Linear Regression

This is a statistical method used to model the relationship between one dependent variable and two or more independent variables. It provides estimates of the coefficients for each explanatory variable, which represent the expected change in the dependent variable associated with a one-unit change in that variable while holding other variables constant. The model's summary typically includes details such as sample size and goodness-of-fit measures like R-squared.

Simple Linear Regression

This technique models the relationship between a dependent variable and a single independent variable. It examines how one predictor alone explains the variation in the outcome and often yields a different slope from a multiple regression, since it does not account for the influence of additional variables that may be affecting the dependent variable.

Logarithmic Transformation

Log transformations are used to linearize relationships, reduce skewness, and stabilize the variance of the variables involved. In regression models, applying a logarithmic transformation to a variable allows for the interpretation of coefficients in terms of percentage changes and can often result in better fit and interpretability of economic data.

Interpretation of Slope Coefficients

The slope coefficients in a regression indicate the magnitude and direction of the relationship between each independent variable and the dependent variable. In models where some variables are logged, the coefficients can be interpreted as the percentage change in the dependent variable for a one percent change in the predictor, provided all other variables remain constant.

Intercept Interpretation

The intercept in a regression model represents the expected value of the dependent variable when all independent variables are set to zero. In cases where it is not meaningful or feasible for the independent variables to be zero (for instance, when logarithms are involved), the intercept may lack a clear substantive interpretation.

Collinearity

Collinearity refers to the situation where two or more explanatory variables in a regression model are highly correlated. This can make it difficult to isolate the individual effect of each variable on the dependent variable and may lead to unstable coefficient estimates. Analyzing the correlation among regressors helps in understanding potential multicollinearity issues.

Model Comparison

Comparing models, such as a simple regression versus a multiple regression, allows us to assess how the inclusion of additional variables may affect the estimation of key coefficients. Discrepancies in estimated coefficients between models can point to omitted variable bias or reveal the extent to which each variable captures variation in the dependent variable.

Transcript

00:02 Hello everyone.

00:03 This is c7 of computer exercises on chapter 3.

00:08 So the first thing we want to do is to estimate the model that mat 10 is equal to beta 0 plus beta 1, log expenditure plus beta 2 lunch program plus you.

00:22 Here, mat 10 is the percentage of students passing the meap mats 10.

00:30 Log expenditure is a log of expenditure and expenditure.

00:33 Is for per students in terms of dollars and lunch program is the percentage of students in school lunch program.

00:44 So first we estimate the model that math -san is equal to constant plus beta 1, log expansion plus beta 2.

00:54 This is the code regress math -sen on log -expandition launch program.

01:01 So the constant beta 0 had is minus 20 .36.

01:12 Beta 1 -1 -6.

01:13 Which is the coefficient on log expenditure is 6 .23 and beta 2 had which is the coefficient on launch program is negative point 305 and r squared from this regression is point 18 are the signs of the slope coefficients what you expect it so let's see beta 1 hat which is the coefficient of log expenditure is positive, which means that spending more per student will increase the percentage of students passing the test.

02:02 This is something we would expect.

02:05 Beta 2 hat has a negative side.

02:11 So this is the coefficient of lunch program, which means as the percentage of students in school lunch program increases, the percentage of students passing the math test decreases.

02:25 That's, well, that could be because it could mean that in some schools where the percentage of students in the last program, it might be that they are spending less.

02:41 There could be some other factors, but this is a little bit weird, maybe i expected.

02:50 Okay, the second question is, what do you make with the intercept? you estimated in part one, in particular, does it make sense to set the two explanations? three variables to zero.

03:01 So beta zero hat in the first part we find is minus 20 .36, which means unlocks expenditure and launch program is equal to zero, then the percentage of students passing the math test is going to be negative 20, which of course doesn't make sense because it should be in between zero or 100.

03:26 But so let's see if it makes sense to set explanatory variables to equal to 0.

03:35 And when we actually look at the data, we see that the minimum that the log expenditure gets is 8.

03:42 Something.

03:44 So it doesn't make sense to set this to be equal to 0.

03:51 And also in the data, we see the minimum that the launch expenditure, launch program percentage is 1 .4.

04:06 So it could make sense to set the large program to equal to zero that like nobody is in school lunch program...

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