Multiple Regression Analysis
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Multiple Regression Analysis. y = b 0 + b 1 x 1 + b 2 x 2 + . . . b k x k + u 4. Further Issues. Redefining Variables.

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Multiple Regression Analysis

y = b0 + b1x1 + b2x2 + . . . bkxk + u

4. Further Issues

Redefining variables
Redefining Variables

  • Suppose we have a model with a variable like income measured in dollars on the left-hand-side. Now we re-define income to be measured in tens of thousands of dollars. What effect will this have on estimation and inference?

  • It will not affect the R2

  • Will such scaling have any effect on t-stats, F-stats and confidence intervals?

    • No, these will also have the same interpretation

  • Changing the scale of the y variable just leads to a corresponding change in the scale of the coefficients

Redefining variables cont
Redefining Variables (cont)

  • Suppose we originally obtain

  • In this specification, house price is measured in dollars.

  • What happens if we re-estimate this with house price measured in thousands of dollars?

Redefining variables cont1
Redefining Variables (cont)

  • If we measure price in thousands of dollars, the new coefficient will be the old coefficient divided by 1000 (same estimated effect!)

  • The standard errors will be 1000 times smaller

  • t-stats etc. will be identical

Redefining variables cont2
Redefining Variables (cont)

  • Changing the scale of one of the x variables:What if we redefine square feet as thousands of square feet? Now all the ’s have the same interpretation as before with the exception of 1-hat

  • It will be 1000 times larger

    • Why? Because now a 1 unit change in square feet is the same as what previously was a 1000 unit change in square feet.

  • The standard error will also be 1000 times larger and t-stats etc. will have the same interpretation

Functional form
Functional Form

  • OLS can be used for relationships that are not strictly linear in x and y by using nonlinear functions of x and y – will still be linear in the parameters


    log(wage)= 0 +1(educ)+2(exper)+3 (exper)2

    In this particular specification we have an example of a log specification with a quadratic term--both are examples of nonlinearities that can be introduced into the standard linear regression model

Interpretation of log models
Interpretation of Log Models

1. If the model is ln(y) = b0 + b1ln(x) + u, then b1 is an elasticity. e.g. if we obtained an estimate of 1.2, this would suggest that a 1 percent increase in x causes y to increase by 1.2 percent.

2. If the model is ln(y) = b0 + b1x + u, then b1*100 is the percent change in y resulting from a unit change in x. e.g. if we obtained an estimate of 0.05, this would suggest that a 1 unit increase in x causes a 5% increase in y.

3. If the model is y = b0 + b1ln(x) + u, then b1/100 is the unit change in y resulting from a 1 percent change in x. e.g. if we obtained an estimate of 20, this would suggest that a 1 percent increase in x causes a 0.2 unit increase in y.

Why use log models
Why use log models?

  • Log models are invariant to the scale of the variables since we’re measuring percent changes

  • They can give a direct estimate of elasticity

  • For models with y > 0, the conditional distribution is often heteroskedastic or skewed, while ln(y) is much less so

  • The distribution of ln(y) is more narrow, limiting the effect of outliers

Some rules of thumb
Some Rules of Thumb

What types of variables are often used in log form?

*Variables in positive dollar amounts

*Variables measuring numbers of people

-school enrollments, population, # employees

*Variables subject to extreme outliers

What types of variables are often used in level form?

*Anything that takes on a negative or zero value

*Variables measured in years

Quadratic models
Quadratic Models

  • Captures increasing or decreasing marginal effects

  • For a model of the form y = b0 + b1x + b2x2 + u, we can’t interpret b1 alone as measuring the change in y with respect to x.

Now the effect of an extra unit of x on y depends in part on the value of x. Suppose b1 is positive. Then if b2 is positive, an extra unit of x has a larger impact on y when x is big than when x is small. If b2 is negative, an extra unit of x has a smaller impact on y when x is big than when x is


More on quadratic models
More on Quadratic Models

  • Suppose that the coefficient on x is positive and the coefficient on x2 is negative

  • Then y is increasing in x at first, but will eventually turn around and be decreasing in x

  • We may want to know the point of inflection

More on quadratic models1
More on Quadratic Models

  • Suppose that the coefficient on x is negative and the coefficient on x2 is positive

  • Then y is decreasing in x at first, but will eventually turn around and be increasing in x

Interaction terms
Interaction Terms

  • We might think that the marginal effect of one RHS variable depends on another RHS variable

    Example: suppose the model can be written: y = b0 + b1x1 + b2x2 + b3x1x2 + u

  • Where y is house price, x1 is the number of square feet and x2 is the number of bedrooms.

  • So the effect of an extra bedroom on price is

Interaction terms1
Interaction Terms

  • If b3>0, this tells us that an extra bedroom boosts the price of a house more, if the square footage of the house is higher.

    • This shouldn’t be surprising. After all, an extra bedroom in a small house is likely to be small compared with an extra bedroom in a large house. So we would expect an extra bedroom in a big house to be worth more.

  • Note that this makes interpretation of b2 a bit less straightforward.

    • Technically, b2 tells us how much an extra bedroom is worth in a house with zero square feet.

    • It may be useful to report on the value of b2+b3x1 for the mean value of x1 .

More on goodness of fit adjusted r squared
More on Goodness-of-Fit: Adjusted R-Squared

  • Recall that the R2 will always increase as more variables are added to the model

  • The adjusted R2 takes into account the number of variables in a model, and may decrease

  • The usual R2 can be written:

Adjusted r squared cont
Adjusted R-Squared (cont)

  • We can define the “population R-squared” as

  • We can use SSR/(n-k-1) as unbiased estimate ofu2

  • Similarly can use SST/(n-1) as unbiased estimate of y2

  • Therefore, adjusted R2 “R-bar squared” is:

Adjusted r squared cont1
Adjusted R-Squared (cont)

  • Notice that R-bar squared can go up or down when a variable is added, unlike the regular R-squared which always goes up

  • R-bar squared is not necessarily “better”- the ratio of 2 unbiased estimators isn’t necessarily unbiased

  • Better to treat it as an alternative way of summarizing goodness of fit

    • If you add a variable to the RHS and the R-bar squared doesn’t rise, this is likely (though not surely) an indication it shouldn’t be included in the model

Comparing nested models
Comparing Nested Models

  • Suppose you wanted to compare the following two models:

    1. y=0+1x+u

    2. y=0+1x+2x2+ u

    We say that (1) is nested in (2); alternatively, (1) is a special case of (2). With a t-test on 2 we can choose between these two models (if reject null of 2=0, we pick model 2). For multiple exclusion restrictions can use F-test.

Comparing non nested models
Comparing Non-Nested Models

  • Suppose you wanted to compare the following two models:

    1. y=0+1log(x)+

    2. y=0+1x+ 2x2+ 

  • One is not nested in the other, so t-test or F-test cannot be used to compare.

  • Here R-bar-squared can be useful. We can simply choose the model with the higher R-bar-squared.

    • Note that a simple comparison of regular R-squared would tend to lead us to choose the model with more explanatory variables.

    • Note that if the LHS variable takes a different form between (1) and (2) we cannot compare using R-bar-squared (or R-squared).

Goodness of fit
Goodness of Fit

  • Important not to fixate too much on adj-R2 and lose sight of theory and common sense

  • If economic theory clearly predicts a variable belongs, generally leave it in

  • Don’t want to exclude a variable that prohibits a sensible interpretation of the variable of interest

  • Remember the ceteris paribus interpretation of multiple regression

Residual analysis
Residual Analysis

  • Sometimes looking at the residuals (i.e. predicted - observed) provides useful information

  • Example: Regress price of cars on characteristics

    • Engine size, efficiency, luxury amenities, roominess, fuel efficiency, etc.

  • Then the residual = actual price - predicted price

    • By picking the car with the lowest (most negative) residual, you would be choosing the most underpriced car (assuming you’re controlling for all relevant characteristics)

Standard errors for predictions
Standard Errors for Predictions

  • Suppose we want to use our estimates to obtain a specific prediction.

  • Such predictions are subject to sampling variation because they are functions of estimated parameters

  • First, suppose that we estimate:


    and we want to obtain a prediction of y for specific value of the x’s. In general we can obtain predictions of y

    by plugging values of the actual x’s into our fitted model.

Standard errors for predictions1
Standard Errors for Predictions

  • Let c1, c2, …, ck denote particular values of the x variables, for which we want to obtain a prediction of y.

  • We can think of estimating a parameter

  • A good estimator is

Predictions cont
Predictions (cont)

  • To get a confidence interval for our estimate of q0 we need its standard error

  • Like testing a linear combination of parameters, the difficulty is getting this standard error

  • Can rewrite as b0 = q0 – b1c1 – … – bkck and plug this into y=0+1x1+…+kxk+u to get

  • If we regress y on a constant and on (x1– c1),…, (xk– ck) we will obtain an estimate of q0 and the standard error of that estimate. This can be used to construct a confidence interval.

Predictions cont1
Predictions (cont)

  • The standard error we obtain here is for the expected value of y given particular x values.. This can be thought of as the standard error of the average y value for the sub-population that has those exact x characteristics. It is not the same as a standard error for a prediction about a particular individual from the population.

  • In order to form a confidence interval for a particular individual we need to also take into account the variance in the unobserved error.

  • Let y0 = some outcome for which we want a confidence interval (for some individual in the population who you wish to make predictions about), x0 = new values of independent variables, and u0 be the unobserved error. Then,

Predictions cont2
Predictions (cont)

  • The best prediction of y0 is:

  • The prediction error is:

  • We know that (because our estimators are unbiased)

Predictions cont3
Predictions (cont)

  • There are 2 sources of variation:

    1. Variance due to the sampling error in prediction (because yhat based on estimated coefficients)

    2. Variance in the error of the population

Prediction interval
Prediction interval

  • We can estimate the standard error of prediction:

Predicting y in a log model
Predicting y in a log model

  • For the prediction

  • Simple exponentiation of the predicted ln(y) will underestimate the expected value of y

  • Use caution when making predictions in a model with ln(y) on LHS (see text pages 219-221).