Refer to Computer Exercise C 2 in Chapter 3 . Now, use the log of the housing price as the depen

Refer to Computer Exercise C 2 in Chapter 3 . Now, use the...

Refer to Computer Exercise $\mathrm{C} 2$ in Chapter $3 .$ Now, use the log of the housing price as the dependent variable: $$\log (price)=\beta_{0}+\beta_{1} \operatorname{sqr} f t+\beta_{2} b d r m s+u$$ (i) You are interested in estimating and obtaining a confidence interval for the percentage change in price when a 150 -square-foot bedroom is added to a house. In decimal form, this is $\theta_{1}=150 \beta_{1}+\beta_{2} .$ Use the data in HPRICE 1 to estimate $\theta_{1}$ . (ii) Write $\beta_{2}$ in terms of $\theta_{1}$ and $\beta_{1}$ and plug this into the log(price) equation. (iii) Use part (ii) to obtain a standard error for $\hat{\theta}_{2}$ and use this standard error to construct a 95$\%$ confidence interval.

Refer to Computer Exercise $\mathrm{C} 2$ in Chapter $3 .$ Now, use the log of the housing price as the dependent variable:
$$\log (price)=\beta_{0}+\beta_{1} \operatorname{sqr} f t+\beta_{2} b d r m s+u$$
(i) You are interested in estimating and obtaining a confidence interval for the percentage change in price when a 150 -square-foot bedroom is added to a house. In decimal form, this is $\theta_{1}=150 \beta_{1}+\beta_{2} .$ Use the data in HPRICE 1 to estimate $\theta_{1}$ .
(ii) Write $\beta_{2}$ in terms of $\theta_{1}$ and $\beta_{1}$ and plug this into the log(price) equation.
(iii) Use part (ii) to obtain a standard error for $\hat{\theta}_{2}$ and use this standard error to construct a 95$\%$ confidence interval.

Key Concepts

Log?Linear Regression

This concept involves using a log-transformed dependent variable in a regression model. It allows interpretation of the coefficients in terms of percentage changes rather than absolute changes, making it particularly useful when the outcome variable exhibits exponential growth or is skewed. The log-linear specification models the logarithm of the dependent variable as a linear function of the predictors.

Interpretation of Regression Coefficients in Log?Linear Models

In log-linear models, the coefficients represent the approximate percentage change in the dependent variable for a one?unit change in the predictor, holding other factors constant. This interpretation is especially useful in economics and other fields where relative changes are more meaningful than absolute differences.

Linear Combination of Regression Coefficients

A linear combination of regression coefficients is a weighted sum of multiple estimated parameters. This concept is used to derive new parameters or test specific hypotheses about the combined effect of changes in the predictors on the transformed dependent variable. It often involves adding or subtracting coefficients after scaling them by relevant factors.

Model Reparameterization

Reparameterization involves expressing one or more parameters of the model in terms of other parameters, often to simplify interpretation or to impose economic theory constraints. This process can lead to alternative model specifications that highlight the effect of interest and facilitate the estimation of standard errors and confidence intervals for transformed parameters.

Standard Error Estimation and Confidence Interval Construction

Standard error estimation quantifies the sampling variability of an estimator, and it is crucial for assessing the precision and reliability of estimates in regression analysis. Confidence intervals are constructed using these standard errors to provide a range in which the true parameter value is likely to fall with a specified probability, such as 95%. Constructing a confidence interval for a linear combination of parameters involves combining the individual variances and covariances according to the weights used in the combination.

Transcript

00:01 Over this problem, you are, the problem asks you to estimate and obtain a confidence interval for the percentage change in price when 150 square foot bedroom is added to a house.

00:14 And you are supposed to estimate an equation that has the log of housing price as a dependent variable.

00:24 So let me just write that out really quick.

00:27 This is your dependent variable.

00:28 And then your independent variables are the square feet of the house and then the number of bedrooms in the house.

00:40 And i'll just write out the estimates you'll get.

00:44 So you'll run that regression as the problem tells you to do with the data.

00:49 And you should get these coefficients and eventually standard errors from the estimation.

01:00 So there's your intercept, and then this is the coefficient on square feet, and finally you have the coefficient on bedrooms.

01:21 So both the coefficient for square feet and bedrooms are positive.

01:24 That makes sense.

01:27 You know, it basically means bigger the houses almost, the more expensive it likely is, the price.

01:36 So there we have the coefficients estimated.

01:37 And then i'll just add in the standard errors too, which we also want to see in blue.

01:48 So here's the standard error for the square foot variable.

01:53 And it looks like it's a lot smaller than the coefficient.

01:57 So the square foot variable seems to be very statistically significant.

02:03 And the bedrooms variable does not look to be nearly as statistically significant.

02:08 The standard error is actually a little bit bigger than the bedroom.

02:12 Coefficient.

02:14 That's what you end up with.

02:15 I'll just add in quickly the sample size as well, which is 88 and the r squared for us is 0 .588.

02:29 So these should be the results you get from the running the regression in your software.

02:36 But this is not the part one is asking you to do exactly.

02:40 They are asking you to obtain a confidence interval or estimate it, start by estimating it, for the percentage change in price when a 150 square foot bedroom is added to the house.

02:57 So what does that look like? first of all, just remember we have this variable, a square foot variable, and we have this bedroom's variable.

03:08 And so what's an expression for? adding a 150 square foot bedroom on.

03:15 So we can just call this theta 1.

03:18 This is adding on the 150 foot bedroom.

03:22 So 150 square foot.

03:29 And then we have to multiply that by beta 1.

03:32 So 150 times the impact of a square foot.

03:35 You can think of it that way.

03:38 And then plus a bedroom.

03:43 So you're adding a bedroom and you're adding 150 square feet is the way to think about it.

03:47 So you're adding a bedroom and you're also adding 150 square feet.

03:54 So this theta 1, once we plug in the numbers that we got for beta 1 and beatty 2, come out to the following numbers.

04:03 You get 150 times beta 1 that we got, which is 0 .000379, and then plus 0 .0289...

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