Multiple Regression Model - Hypothesis Testing

EC295

Justin Smith

Wilfrid Laurier University

Fall 2022

Introduction

Introduction

  • Recall that we can use estimates to test hypotheses about slope parameters

  • In this section, we learn how to do this in a multiple regression

  • In simple regression, hypotheses tests are about one parameter

  • In multiple regression we can do more

    • Tests of single hypotheses

      • Involving one parameter

      • Or multiple parameters

    • Tests of multiple hypotheses

      • Can jointly evaluate more than one hypothesis

  • We will also learn confidence intervals for multiple parameters

    • Called a confidence set

Introduction

  • Finally, we will learn about model specification

    • Choosing which variables to include in the regression

    • In many situations, we are interested in an unbiased estimate of a single parameter

      • e.g. effect of class size on test scores

    • In that case, other variables are added to “control” omitted variables bias

      • Control variables are not object of interest

      • Added to model to get unbiased estimate of main parameter

    • We will learn how to get causal estimate of key parameter even when control variables are related to the error

      • Change assumption about how error is related to x

Testing Hypotheses about Single Parameters

Hypothesis Tests for a Single Parameters

  • Tests about single parameters in multiple regression follows same process as single regression

  • Use a t-test

    1. Formulate opposing hypotheses about \(\beta_{j}\)

    2. Choose test statistic

    3. Derive a decision rule

    4. Use sample data to compute the observed value of the test statistic and confront it with the decision rule

  • Or, use p-value in place of 3 and 4

  • Opposing hypotheses for two-sided test are

    • \(H_{0}:\beta_{j} = \beta_{j,0}\)

    • \(H_{1}: \beta_{j} \ne \beta_{j,0}\)

Hypothesis Tests for a Single Parameters

  • Test statistic is again the t-statistic

    \[t = \frac{\hat{\beta}_{j} - \beta_{j,0}}{se(\hat{\beta}_{j})}\]

    • \(\beta_{j,0}\) is value of the null hypothesis

  • Decision rule

    • Reject null if estimate \(\hat{\beta}_{j}\) is too far away from the claim

    • Set by the significance level, and associated critical value

  • Can instead use p-value

    • For 2-sided test p-value \(= 2Pr(t > |t^{act}|)\)

Standard Errors for OLS Estimators

  • Recall the standard error is the estimate of the standard deviation of \(\hat{\beta}_{j}\)

  • Necessary to compute the t-statistic

  • In simple regression, we derived formulas for \(se(\hat{\beta}_{j})\)

    • Under heteroskedasticity and homoskedasticity

  • In multiple regression, formulas are more complex

    • Under homoskedasticity \[\hat{Var}[\hat{\beta}_{j}|X_{1i},X_{2i},...,X_{ki}] = \frac{\frac{1}{n-k-1}\sum_{i=1}^{n}\hat{u}_{i}^2 }{(\sum_{i=1}^{n}(X_{ji} - \bar{X}_{j})^2)(1-R^2_{j})}\]

    • Under heteroskedasticity formula is messier, but same intuition

Standard Errors for OLS Estimators

  • For testing hypotheses, \(se(\hat{\beta}_{j})\) are computed using software

    • Remember: in Stata, use “robust” option to get heteroskedasticity-robust standard errors

  • Use these in exactly the same way we did before

Confidence Intervals for Single Coefficients

  • A confidence interval for \(\beta_{j}\) is computed as \[\hat{\beta}_{j} \pm t^c \times se(\hat{\beta}_{j})\]

    • Recall that if computing \((1-\alpha)\%\) interval, then \(t^c\) is critical value from 2-sided \(\alpha\%\) hypothesis test

  • Constructing this interval is the same as for simple regression

Example with Stata

  • Research Question: What are the factors that determine birth weight?

  • Low birth weight is linked to many negative future outcomes

    • Health

    • Developmental

    • Economic

  • Smoking while pregnant is identified as a major cause of low birth weight

  • It is therefore important to curb maternal smoking

Example with Stata

  • Suppose the model explaining birth weight is \[bwghtlbs= \beta_{0} + \beta_{1}packs + \beta_{2}motheduc + \beta_{3} fatheduc + \beta_{4} faminc+ u\]

    • \(\beta_{1}\) is ceteris paribus effect of one extra pack of cigarettes smoked by the mother per day

    • \(\beta_{2}\) is ceteris paribus effect of a 1-year increase in father education

    • \(\beta_{3}\) is ceteris paribus effect of a 1-year increase in mother education

    • \(\beta_{4}\) is ceteris paribus effect of a 1000 increase in family income

    • \(\beta_{0}\) is birth weight when all variables and the error are zero

    • \(u\) are unobserved factors that explain birth weight

Example with Stata

  • We have information on 1,388 children and their families

    • Birth weight

    • Family income and education

    • Maternal smoking

    • Tobacco prices and taxes

  • Drop a few values due to missing information on education

  • The datafile and dofile are posted to mylearningspace

    • Datafile: bwght.dta

    • Dofile: EC295 Inference Example

Example with Stata

use "bwght.dta", clear
keep bwghtlbs packs motheduc fatheduc faminc
sum bwghtlbs packs motheduc fatheduc faminc
    Variable |        Obs        Mean    Std. dev.       Min        Max
-------------+---------------------------------------------------------
    bwghtlbs |      1,388    7.418723    1.272123     1.4375    16.9375
       packs |      1,388    .1043588    .2986344          0        2.5
    motheduc |      1,387    12.93583    2.376728          2         18
    fatheduc |      1,192    13.18624    2.745985          1         18
      faminc |      1,388    29.02666    18.73928         .5         65

(196 observations deleted)

(1 observation deleted)

  • There are some missing values for motheduc and fatheduc

    • We will drop these

  • Note that famic is measured in thousands

Example with Stata

  • Estimating the parameters by OLS
regress bwghtlbs packs motheduc fatheduc faminc, robust
Linear regression                               Number of obs     =      1,191
                                                F(4, 1186)        =      11.47
                                                Prob > F          =     0.0000
                                                R-squared         =     0.0328
                                                Root MSE          =     1.2401

------------------------------------------------------------------------------
             |               Robust
    bwghtlbs | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       packs |  -.7368692   .1303338    -5.65   0.000    -.9925797   -.4811587
    motheduc |  -.0273702   .0184606    -1.48   0.138    -.0635892    .0088488
    fatheduc |   .0308543   .0165627     1.86   0.063     -.001641    .0633497
      faminc |   .0033641    .002215     1.52   0.129    -.0009817    .0077099
       _cons |   7.379629   .1995718    36.98   0.000     6.988075    7.771182
------------------------------------------------------------------------------

  • Smoking has the expected negative effect on birth weight

  • Mother and father education have opposite effects

    • Not clear why

  • All else equal, family income has a small effect

Example with Stata

  • Without any more calculation, we can test \(H_{0}: \beta_{j} = 0\) vs \(H_{1}: \beta_{j} \neq 0\)

    • Stata automatically reports this beside slope estimates

  • Using the p-values in our example, at the 5% level

    • packs is statistically significant

    • All other variables are not statistically significant at that level

    • Recall that to reject the null, p-value \(\le \alpha\)

  • We test other null hypotheses also, but we need to do calculations

Example with Stata

  • Suppose we want to test \(H_{0}: \beta_{1} = -1\) vs \(H_{1}: \beta_{1} \neq -1\)

    • The observed value of the test statistic is

      \[t = \frac{\hat{\beta}_{j} - \beta_{j,0}}{se(\hat{\beta}_{j})} = \frac{ -.736892 + 1}{.1303338} = 2.0187\]

    • \(\hat{\beta}_{j}\) is 2.0187 standard deviations above hypothesized value

    • The critical value at the 5% level is \(1.96\)

      • Degrees of freedom \(=n-k-1 = 1191 - 4 - 1 = 1186\)

      • stata code: display invttail(1186,.025)

    • We fail to reject \(H_{0}\) in this case

    • P-value is 0.0437

      • stata code: display 2*ttail(1186,2.0187)

      • Reject if \(\alpha \ge 0.0437\)

Example with Stata

  • Regression table has 95% confidence intervals for each \(\hat{\beta}_{j}\)

    • For packs, accept any \(H_{0}\) between -.9925797 and -.4811587

    • Any confidence interval that includes 0 means \(\hat{\beta}_{j}\) is statistically insignificant

Tests of Joint Hypotheses

Introduction

  • So far, we have only tested single hypotheses involving one parameter

    • e.g. is \(\beta_{j}\) statistically different from zero?

  • In a multiple regression, we might be interested in more than one test at a time

    • Ex: are \(\beta_{1}\) and \(\beta_{2}\) both statistically different from zero?

    • Ex: are \(\beta_{1}\) and \(\beta_{2}\) both statistically different from each other?

    • Ex: are all parameters in the regression different from zero?

  • When testing more than one hypothesis at a time, we need to alter the procedure slightly

  • Biggest change is we use a different test statistic

    • Use F-statistic for these tests

Hypotheses about Two or More Parameters

  • Suppose we are interested in the test score regression \[TestScore_{i} = \beta_{0} + \beta_{1}STR_{i} + \beta_{2}Expn_{i} + \beta_{3}PctEL_{i} + u_{i}\]

    • STR is student-teacher ratio

    • Expn is expenditures per student

    • PctEL is fraction in ESL

  • Test whether class size and student expenditures have an effect on test scores

  • We can test this with a joint null hypothesis \[H_{0}: \beta_{1}=0 \text{ and } \beta_{2}=0\] \[H_{1}: \beta_{1}\neq 0 \text{ and/or } \beta_{2}\neq 0\]

Hypotheses about Two or More Parameters

  • Null is that both parameters are zero

    • Imposes two restrictions on the values of the parameters

    • Testing two hypotheses at the same time

  • Alternative is one or both parameters are non-zero

    • Null and alternative hypotheses have to account for all possible values of parameters

    • Opposite to both parameters being zero is one or both not zero

  • In a more general model \[Y_{i} = \beta_{0} + \beta_{1}X_{1i} + \beta_{2}X_{2i}+ ... + \beta_{k}X_{ki} + u_{i}\]

  • Imagine we want to test two or more restrictions (hypotheses)

    • Call the number of restrictions \(q\)

Hypotheses about Two or More Parameters

  • A joint hypothesis tests \(q\) restrictions on the parameters \[H_{0}: \beta_{j}=\beta_{j,0},\beta_{m}=\beta_{m,0}, ... \text{ for a total of q restrictions }\] \[H_{1}:\text{ One or more of the q restrictions under } H_{0} \text { does not hold}\]

  • Examples

    • \(H_{0}: \beta_{1} = 0, \beta_{2} = 0, \beta_{4} = 0\); \(H_{1}:\) one or all is not zero

      • Testing \(q=3\) total hypotheses

    • \(H_{0}: \beta_{2} = 4, \beta_{5} = -2\); \(H_{1}:\) one or both not those values

      • Testing \(q=2\) total hypotheses

  • The appropriate test statistic for joint tests is the F-statistic

    • We cover this statistic in detail below

Why not test hypotheses individually?

  • Suppose again that model is \[TestScore_{i} = \beta_{0} + \beta_{1}STR_{i} + \beta_{2}Expn_{i} + \beta_{3}PctEL_{i} + u_{i}\]

  • Test whether class size and student expenditures have an effect on test scores

  • The null and alternative hypotheses are \[H_{0}: \beta_{1}=0 \text{ and } \beta_{2}=0\] \[H_{1}: \beta_{1}\neq 0 \text{ and/or } \beta_{2}\neq 0\]

  • Suppose we test this hypothesis with individual t-tests

    • \(t_{1}\) is the t-statistic for testing \(\beta_{1} = 0\)

    • \(t_{2}\) is the t-statistic for testing \(\beta_{2} = 0\)

Why not test hypotheses individually?

  • Strategy is reject \(H_{0}\) if either \(t_{1}\) or \(t_{2}\) exceeds critical value

  • Suppose we want 5% significance level

    • Probability of making Type I error is 5%

    • Critical value is 1.96 with large sample

  • Problem with strategy above is it produces Type I error rate larger than \(5\%\)

  • To see this, assume estimators of \(\beta_{1}\) and \(\beta_{2 }\) are unrelated

    • Accept \(H_{0}\) when \(t_{1} < 1.96\) and \(t_{2} < 1.96\)

    • The likelihood of this happening when \(H_{0}\) is true is \[Pr[t_{1} < 1.96, t_{2} < 1.96] = Pr[t_{1} < 1.96] \times Pr[t_{2} < 1.96]\] \[= 0.95 \times 0.95 = 0.9025\]

Why not test hypotheses individually?

  • The probability of rejecting \(H_{0}\) when it is true is therefore \[1 - Pr[t_{1} < 1.96, t_{2} < 1.96] =1-0.9025 = 0.0975\]

  • So with individual t-test strategy, Type I error rate is actually 9.75%

  • Why?

    • You get multiple opportunities to reject null

      • If don’t reject using first t-test, you can reject using second

      • Effectively, you get two “kicks at the can”

    • Drives up probability of rejecting \(H_{0}\) when it is true

  • For this reason, we cannot use individual testing approach

The F-Statistic with 2 Restrictions

  • The appropriate statistic for testing multiple hypotheses is the F-statistic

    • When testing with F-statistic, we call it an F-test

  • With only 2 restrictions and testing \(H_{0}: \beta_{1}=0 \text{ and } \beta_{2}=0\)

    \[F = \frac{1}{2} \left ( \frac{t_{1}^2 + t_{2}^2 - 2\hat{\rho}_{t_{1},t_{2}}t_{1}t_{2}}{1-\hat{\rho}_{t_{1},t_{2}}^2} \right )\]

  • In this formula

    • \(t_{1}\) is the t-statistic for testing \(\beta_{1} = 0\)

    • \(t_{2}\) is the t-statistic for testing \(\beta_{2} = 0\)

    • \(\hat{\rho}_{t_{1},t_{2}}\) is the correlation between the two t-statistics

  • Formula valid for heteroskedastic or homoskedastic errors

The F-Statistic with 2 Restrictions

  • In the F-test, we will reject \(H_{0}\) when the F-statistic is large

  • To understand intuition, imagine t-statistics are uncorrelated

    • Set \(\hat{\rho}_{t_{1},t_{2}} = 0\)

    • In this case \(F = \frac{1}{2} \left( t_{1}^2 + t_{2}^2 \right )\)

    • F-stat is big when either \(t_{1}\) or \(t_{2}\) or both is big

      • When either hypothesis is very different from the claim

    • This will lead us to reject the null that both slopes are zero

    • More complicated intuition when \(\hat{\rho}_{t_{1},t_{2}} \neq 0\)

  • Whether the F-statistic is large enough is determined by

    • The sampling distribution of the F-statistic

    • The significance level of the test

The F-Statistic with 2 Restrictions

  • This F-statistic has an F distribution

    • The F-distribution is big mass at low values and long right tail

    • Only defined over positive values

    • Depends on two sets of degrees of freedom

      • \(q\), the number of restrictions

      • \(n-k-1\), the model degrees of freedom

    • Graph on next slide shows F distribution with different numbers of restrictions and 1000 model degrees of freedom

  • Use this distribution to get critical values, p-values for the F-test

    • Analogous to t-testing, with different test statistic and sampling distribution

Testing Multiple Linear Restrictions

The F-Statistic with q Restrictions

  • With more than 2 restrictions, heteroskedasticity-robust formula is more complicated

    • Cannot write it down easily without matrix algebra

  • Formula is computed automatically in Stata (and other software)

  • Intuition is the same

    • F-statistic has an F-distribution with q, n-k-1 degrees of freedom

    • Reject \(H_{0}\) when F-stat is large

    • With significance level, obtain critical value from F-distribution

    • Or compute p-values based on F-statistic

P-values in an F-test

  • Like in t-testing, we can compute a p-value for F-tests

  • It is probability of getting F-stat larger than the one we observe \[p-value = Pr[F_{q,n-k-1} > F^{act}]\]

  • Like before, we need Stata to compute this p-value

    • Use “Ftail” function

  • Intuition of p-value is the same

    • Reject \(H_{0}\) for all \(\alpha\) larger than p-value

F-test of Overall Significance

  • Common to test whether any variable in the regression is significant

    \[H_{0}: \beta_{1}=0, \beta_{2}=0,..., \beta_{k}=0\] \[H_{1}: \text{Any one or more of } \beta_{1}, \beta_{2}, ...,\beta_{k} \text{ is not zero}\]

  • Called test of overall significance because we are testing whether any variable matters

    • Accepting null means none of them matter

    • Rejecting null means at least one matters

  • This test statistic is automatically reported in regression output

    • In top right corner of Stata regression table

The F-Statistic with 1 Restriction

  • Finally, you can use an F-test test single hypotheses

    • We have been using t-tests for this

  • When testing one restriction, then

    \[F = t^2\]

  • An F-test and t-test with one restriction will produce identical conclusions

    • Given direct link between the test statistics

    • So, you can use either one

Example with Stata

  • Recall the birthweight example

  • The regression model we are estimating is \[bwghtlbs= \beta_{0} + \beta_{1}packs + \beta_{2}motheduc + \beta_{3} fatheduc + \beta_{4} faminc+ u\]

  • Suppose we want to test the joint hypotheses \[H_{0}: \beta_{2}= 0, \beta_{3}=0, \beta_{4}=0\] \[H_{1}: \mbox{Any one or more of } \beta_{2}, \beta_{3}, \beta_{4} \mbox{ is not zero}\]

  • In this example we have q=3 restrictions

  • To get the heteroskedasticity-robust F-statistic, we need Stata

  • The command for F-tests following regression is “test”

Example with Stata

regress bwghtlbs packs motheduc fatheduc faminc, robust
test motheduc fatheduc faminc
Linear regression                               Number of obs     =      1,191
                                                F(4, 1186)        =      11.47
                                                Prob > F          =     0.0000
                                                R-squared         =     0.0328
                                                Root MSE          =     1.2401

------------------------------------------------------------------------------
             |               Robust
    bwghtlbs | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       packs |  -.7368692   .1303338    -5.65   0.000    -.9925797   -.4811587
    motheduc |  -.0273702   .0184606    -1.48   0.138    -.0635892    .0088488
    fatheduc |   .0308543   .0165627     1.86   0.063     -.001641    .0633497
      faminc |   .0033641    .002215     1.52   0.129    -.0009817    .0077099
       _cons |   7.379629   .1995718    36.98   0.000     6.988075    7.771182
------------------------------------------------------------------------------


 ( 1)  motheduc = 0
 ( 2)  fatheduc = 0
 ( 3)  faminc = 0

       F(  3,  1186) =    2.66
            Prob > F =    0.0469

Example with Stata

  • The test command does the F-test for multiple restrictions

    • Reminder: this is not the same as the ttest command

  • The output reports the following things

    • The restrictions it is testing

    • The F-statistic

    • The p-value for the F-test

  • If you use the robust option in your regression, the F-stat in “test” will be heteroskedasticity-robust

    • If you do not use the robust option, it will report the homoskedasticity-only F-stat

  • Critical value from F(3,1186) distribution and \(\alpha = 0.05\)

    • Stata code: display invFtail(3,1186,0.05)

    • \(F^c = 2.61\)

Homoskedasticity-only F-statistic

  • The F-statistics discussed above are robust to heteroskedasticity

    • You can use them with both heteroskedastic and homoskedastic errors

  • If you are sure your errors are homoskedastic, we can use alternate formula

  • This formula is based on the sum of squared residuals (SSR) from two regressions

    1. Unrestricted model: the original regression model estimated with all variables

    2. Restricted model: the model after restrictions in \(H_{0}\) are imposed

Homoskedasticity-only F-statistic

  • The homoskedasticity-only F-statistic is \[F = \frac{(SSR_{r} - SSR_{ur})/q }{SSR_{ur}/(n-k-1)}\]

  • Test is based on changes in SSR when we impose restrictions

    • SSR always increases when variables are excluded from a model

      • Equivalently, \(SSR_{r} \ge SSR_{ur}\)

      • With fewer variables in model, more variation in residual

    • The F-test is based on how much SSR increases when we exclude variables

    • F-stat will be large when \(SSR_{r}\) is much higher than \(SSR_{ur}\)

      • Happens if imposing restriction significantly worsens model fit

      • This will lead us to reject the restrictions in \(H_{0}\)

Homoskedasticity-only F-statistic

  • You can also express the homoskedasticity-only F-statistic in terms of \(R^2\) \[F = \frac{(R^2_{ur} - R^2_{r})/q }{(1-R^2_{ur})/(n-k-1)}\]

  • We learned previously that \(R^2\) rises with more variables

    • So, \(R^2_{ur} \ge R^2_{r}\)

    • Model with more variables has higher \(R^2\) than model with less

  • F-stat will be large if \(R^2\) rises a lot when we remove restrictions

    • Happens when adding variables significantly increases fit

    • This will lead us to reject the restrictions in \(H_{0}\)

  • Both versions of this test statistic say the same thing

    • If restrictions significantly worsen fit, we reject them

Homoskedasticity-only F-statistic

  • Return again to birthweight example

  • Unrestricted model is \[bwghtlbs= \beta_{0} + \beta_{1}packs + \beta_{2}motheduc + \beta_{3} fatheduc + \beta_{4} faminc+ u\]

  • Hypotheses \[H_{0}: \beta_{2}= 0, \beta_{3}=0, \beta_{4}=0\] \[H_{1}: \mbox{Any one of } \beta_{2}, \beta_{3}, \beta_{4} \mbox{ is not zero}\]

  • Restricted model (when we impose restrictions in \(H_{0}\)) \[bwghtlbs= \beta_{0} + \beta_{1}packs + u\]

Example with Stata

*unrestricted model
regress bwghtlbs packs motheduc fatheduc faminc
      Source |       SS           df       MS      Number of obs   =     1,191
-------------+----------------------------------   F(4, 1186)      =     10.05
       Model |  61.8267942         4  15.4566986   Prob > F        =    0.0000
    Residual |  1823.90247     1,186  1.53786043   R-squared       =    0.0328
-------------+----------------------------------   Adj R-squared   =    0.0295
       Total |  1885.72927     1,190  1.58464644   Root MSE        =    1.2401

------------------------------------------------------------------------------
    bwghtlbs | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       packs |  -.7368692   .1382715    -5.33   0.000    -1.008153   -.4655852
    motheduc |  -.0273702   .0199836    -1.37   0.171    -.0665774    .0118369
    fatheduc |   .0308543   .0177056     1.74   0.082    -.0038834    .0655921
      faminc |   .0033641   .0022906     1.47   0.142    -.0011301    .0078582
       _cons |   7.379629   .2187682    33.73   0.000     6.950413    7.808844
------------------------------------------------------------------------------

Example with Stata

*restricted model
regress bwghtlbs packs
      Source |       SS           df       MS      Number of obs   =     1,191
-------------+----------------------------------   F(1, 1189)      =     33.10
       Model |  51.0775373         1  51.0775373   Prob > F        =    0.0000
    Residual |  1834.65173     1,189   1.5430208   R-squared       =    0.0271
-------------+----------------------------------   Adj R-squared   =    0.0263
       Total |  1885.72927     1,190  1.58464644   Root MSE        =    1.2422

------------------------------------------------------------------------------
    bwghtlbs | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       packs |  -.7753962   .1347704    -5.75   0.000    -1.039811   -.5109819
       _cons |   7.539201   .0379168   198.84   0.000     7.464809    7.613592
------------------------------------------------------------------------------
*Critical value
display invFtail(3,1186,0.05)
2.6124063

Example with Stata

  • From the output, we can obtain

    • \(SSR_{ur} = 1823.90247\)

    • \(SSR_{r} = 1834.65173\)

    • \(q =3\)

    • \(n-k-1 = 1186\)

  • Bringing this together \[F = \frac{(SSR_{r} - SSR_{ur})/q}{SSR_{ur}/(n-k-1)} =\frac{(1834.65173 - 1823.90247)/3}{1823.90247/1186} = 2.33\]

  • Critical value from F(3,1186) distribution and \(\alpha = 0.05\)

    • Stata code: display invFtail(3,1186,0.05)

    • \(F^c = 2.61\)

Example with Stata

  • Using the \(R^2\) version of the formula

    • \(R^2_{ur} = 0.0328\)

    • \(R^2_{r} = 0.0271\)

    • \(q =3\)

    • \(n-k-1 = 1186\)

  • Bringing this together \[F = \frac{(R^2_{ur} - R^2_{r})/q }{(1-R^2_{ur})/(n-k-1)} =\frac{(0.0328- 0.0271)/3}{(1-0.0328)/1186} = 2.33\]

  • Critical value from F(3,1186) distribution and \(\alpha = 0.05\)

    • Stata code: display invFtail(3,1186,0.05)

    • \(F^c = 2.61\)

Example with Stata

  • Can also just use “test” command

  • Note how F-statistic is different from earlier

    • Because we did not specify “robust” in the regression
regress bwghtlbs packs motheduc fatheduc faminc
test motheduc fatheduc faminc
      Source |       SS           df       MS      Number of obs   =     1,191
-------------+----------------------------------   F(4, 1186)      =     10.05
       Model |  61.8267942         4  15.4566986   Prob > F        =    0.0000
    Residual |  1823.90247     1,186  1.53786043   R-squared       =    0.0328
-------------+----------------------------------   Adj R-squared   =    0.0295
       Total |  1885.72927     1,190  1.58464644   Root MSE        =    1.2401

------------------------------------------------------------------------------
    bwghtlbs | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       packs |  -.7368692   .1382715    -5.33   0.000    -1.008153   -.4655852
    motheduc |  -.0273702   .0199836    -1.37   0.171    -.0665774    .0118369
    fatheduc |   .0308543   .0177056     1.74   0.082    -.0038834    .0655921
      faminc |   .0033641   .0022906     1.47   0.142    -.0011301    .0078582
       _cons |   7.379629   .2187682    33.73   0.000     6.950413    7.808844
------------------------------------------------------------------------------


 ( 1)  motheduc = 0
 ( 2)  fatheduc = 0
 ( 3)  faminc = 0

       F(  3,  1186) =    2.33
            Prob > F =    0.0728

Testing Hypotheses About Combinations of Parameters

Introduction

  • So far, we have tested restrictions that involve one parameter each

    • e.g. \(H_{0}: \beta_{1} = 0\)

    • e.g. \(H_{0}: \beta_{1} = 0, \beta_{2} = 0\)

  • You can also test hypotheses involving multiple parameters

  • Suppose again the model explaining birth weight is \[bwghtlbs= \beta_{0} + \beta_{1}packs + \beta_{2}motheduc + \beta_{3} fatheduc + \beta_{4} faminc+ u\]

  • And we want to know whether mother and father education have the same effect on birth weight

  • The null and alternative hypotheses are

    • \(H_{0}: \beta_{2} = \beta_{3}\)

    • \(H_{1}: \beta_{2} \neq \beta_{3}\)

Introduction

  • Reject if there is a big difference in education effects

    • Equivalently, reject if difference is much higher or much lower than zero

    • Can rewrite null and alternative hypotheses as

      • \(H_{0}: \beta_{2} - \beta_{3} = 0\)

      • \(H_{1}: \beta_{2} - \beta_{3}\neq 0\)

  • How do we conduct this test?

Approach 1: Use t-or F-test

  • This is a situation with q=1 restrictions

  • You can use a regular t- or F-test

  • The t-statistic for such a test would be

    \[t = \frac{\hat{\beta}_{3} - \hat{\beta}_{2}}{se(\hat{\beta}_{3} - \hat{\beta}_{2})}\]

  • It is difficult to compute \(se(\hat{\beta}_{3} - \hat{\beta}_{2})\)

    • Remember rules of variance \[Var(\hat{\beta}_{3} - \hat{\beta}_{2}) = Var(\hat{\beta}_{3}) + Var(\hat{\beta}_{2}) - 2Cov(\hat{\beta}_{3}, \hat{\beta}_{2})\] \[se(\hat{\beta}_{3} - \hat{\beta}_{2}) = \sqrt{se(\hat{\beta}_{3})^2 + se(\hat{\beta}_{2})^2 -2Cov(\hat{\beta}_{3}, \hat{\beta}_{2})}\]

Approach 1: Use t-or F-test

  • We do not usually have \(Cov(\hat{\beta}_{3}, \hat{\beta}_{2})\) available

    • Need Stata or other software to compute this

  • You can also use an F-test

    • This is a situation with q=1 restrictions

  • The “test” command in Stata computes the F-statistic

Example with Stata

regress bwghtlbs packs motheduc fatheduc faminc, robust
test motheduc= fatheduc
Linear regression                               Number of obs     =      1,191
                                                F(4, 1186)        =      11.47
                                                Prob > F          =     0.0000
                                                R-squared         =     0.0328
                                                Root MSE          =     1.2401

------------------------------------------------------------------------------
             |               Robust
    bwghtlbs | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       packs |  -.7368692   .1303338    -5.65   0.000    -.9925797   -.4811587
    motheduc |  -.0273702   .0184606    -1.48   0.138    -.0635892    .0088488
    fatheduc |   .0308543   .0165627     1.86   0.063     -.001641    .0633497
      faminc |   .0033641    .002215     1.52   0.129    -.0009817    .0077099
       _cons |   7.379629   .1995718    36.98   0.000     6.988075    7.771182
------------------------------------------------------------------------------


 ( 1)  motheduc - fatheduc = 0

       F(  1,  1186) =    3.53
            Prob > F =    0.0605

Approach 2: Use modified regression

  • We can also implement the test with a modified regression

  • Define the parameter \(\theta\) as \[\theta = \beta_{3} - \beta_{2}\]

  • Then we can recast the hypotheses as

    • \(H_{0}: \theta = 0\)

    • \(H_{1}: \theta \neq 0\)

  • So now we are testing hypotheses about \(\theta\)

  • We just need to find a way to estimate \(\theta\)

  • First, rearrange the \(\theta\) equation as \[\beta_{3} = \beta_{2} + \theta\]

Approach 2: Use modified regression

  • Then substitute into the regression \[bwghtlbs= \beta_{0} + \beta_{1}packs + \beta_{2}motheduc\] \[+ (\beta_{2} + \theta) fatheduc + \beta_{4} faminc+ u\]

  • Rearranging \[bwghtlbs= \beta_{0} + \beta_{1}packs + \beta_{2}(motheduc + fatheduc)\] \[+ \theta fatheduc + \beta_{4} faminc+ u\]

  • Based on this, we can estimate \(\theta\) by estimating \[bwghtlbs= \beta_{0} + \beta_{1}packs + \beta_{2}toteduc + \theta fatheduc + \beta_{4} faminc+ u\]

    • where \(toteduc =motheduc + fatheduc\)

    • You need to create the variable \(toteduc\) before estimating the regression

Approach 2: Use modified regression

  • When you estimate the regression above, it will produce the estimate \[\hat{\theta} = \hat{\beta}_{3} - \hat{\beta}_{2}\]

  • The standard error estimate for \(\hat{\theta}\) will therefore measure \[se(\hat{\theta}) = se(\hat{\beta}_{3} - \hat{\beta}_{2})\]

  • You can then do a regular t-test of \(H_{0} : \theta = 0\)

    • In this example, test slope on \(fatheduc\)

    • Not coefficient on \(toteduc\)

  • Next slide implements this test with our Stata example

Example with Stata

gen toteduc = motheduc + fatheduc
regress bwghtlbs packs toteduc fatheduc faminc, robust
Linear regression                               Number of obs     =      1,191
                                                F(4, 1186)        =      11.47
                                                Prob > F          =     0.0000
                                                R-squared         =     0.0328
                                                Root MSE          =     1.2401

------------------------------------------------------------------------------
             |               Robust
    bwghtlbs | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       packs |  -.7368692   .1303338    -5.65   0.000    -.9925797   -.4811587
     toteduc |  -.0273702   .0184606    -1.48   0.138    -.0635892    .0088488
    fatheduc |   .0582246   .0309845     1.88   0.060     -.002566    .1190151
      faminc |   .0033641    .002215     1.52   0.129    -.0009817    .0077099
       _cons |   7.379629   .1995718    36.98   0.000     6.988075    7.771182
------------------------------------------------------------------------------

Confidence Sets for Multiple Coefficients

Confidence Sets

  • We have learned several times about confidence intervals for single parameters

    • An interval estimate for a parameter

    • Parameter is inside interval in \((1-\alpha)\%\) of samples

    • Values inside interval are accepted in 2-sided t-test at \(\alpha\%\) level

  • The concept for multiple parameters is the confidence set

    • Confidence Set: the set that contains the parameters in \((1-\alpha)\%\) of repeated samples

  • This is a generalization of the confidence interval for 2 or more parameters

  • Formula is more complicated, so we will not show it

    • It is based on the F-statistic we learned previously

Confidence Sets

  • Like before, we can use confidence sets for hypothesis testing

    • Suppose we construct a \((1-\alpha)\%\) confidence set

    • Any combination of parameter values in set is accepted in an F-test at \(\alpha\%\) level

    • Any combination outside the set is rejected

  • Ex: Suppose we are again estimating the regression \[TestScore_{i} = \beta_{0} + \beta_{1}STR_{i} + \beta_{2}Expn_{i} + \beta_{3}PctEL_{i} + u_{i}\]

  • A 95% confidence set for the parameters \(\beta_{1}\) and \(\beta_{2}\) is shown on next slide

    • Graphs \(\beta_{2}\) against \(\beta_{1}\)

    • Shaded ellipse shows confidence set

    • Center of set is estimates \(\hat{\beta}_{1}\) and \(\hat{\beta}_{2}\)

Confidence Sets

Confidence Sets

  • Any parameter combination inside ellipse is accepted

  • Confidence set takes elliptical shape with two parameters

    • Cannot plot graph with \(>2\) parameters

  • Ellipse tilts rightward because \(\hat{\beta}_{1}\) and \(\hat{\beta}_{2}\) positively correlated

    • If negatively correlated, it would tilt leftward

    • If uncorrelated, it would not tilt

    • Correlations between \(\hat{\beta}_{1}\) and \(\hat{\beta}_{2}\) driven by correlations in underlying variables

  • Higher standard errors make ellipse taller, wider

    • Sampling variation in \(\hat{\beta}_{1}\) makes it wider

    • Sampling variation in \(\hat{\beta}_{2}\) makes it taller

Model Specification

Introduction

  • Model specification refers to what variables enter a model

  • This is trickiest aspect of applied econometrics

  • Main goal of many econometric analyses is the unbiased effect of a single independent variable on \(Y\)

    • Effect of class size on test scores

    • Effect of education on wages

  • In this case, additional variables only added to model to avoid omitted variables bias

    • Include variable if related to \(Y\) and main \(X\) of interest

    • These variables are called “control variables”

  • So a good starting point for model specification is avoidance of omitted variables bias

Introduction

  • Requires giving thought to determinants of \(Y\)

    • Some background knowledge of research question is important

    • E.g. need to know that wealthier school districts have smaller classes and do better on tests

  • You should avoid using only statistical measures of fit as reason to add/remove variable

    • Sometimes people use \(R^2\) or \(\bar{R}^2\)

    • They say only whether \(X\) variables are good at explaining \(Y\) in sample at hand

    • It is uninformative about causality or omitted variables

    • Can have high \(R^2\) but biased estimates

Control Variables in Multiple Regression

  • As mentioned above, many econometric analyses seek an unbiased effect of a single independent variable on \(Y\)

  • So we differentiate between two types of independent variables

    • Variable of interest: regressor for which we want to estimate a causal effect

    • Control variable: regressor included to hold constant factors that would lead to omitted variables bias if neglected. Not the object of interest

  • Our initial OLS assumptions did not make this distinction

    • Treated all variables like variables of interest

    • So we assumed that all \(X\) variables were uncorrelated with \(u\)

Control Variables in Multiple Regression

  • Most of the time, we do not need an unbiased estimate for control variables

  • This means we can loosen OLS assumptions

  • Suppose regression model is \[Y_{i} = \beta_{0} + \beta_{1}X_{1i} + \beta_{2}X_{2i} + u_{i}\]

  • Imagine \(X_{1i}\) is variable of interest

    • We want an unbiased estimate of \(\beta_{1}\)

  • \(X_{2i}\) is a control variable

    • We do not need an unbiased estimate of \(\beta_{2}\)

  • In this scenario, we do not have to assume \(E[u_{i}|X_{1i}, X_{2i}] = 0\)

    • Do not need both variables to be unrelated to \(u_{i}\)

Control Variables in Multiple Regression

  • Instead, we can assume conditional mean independence

    • Conditional Mean Independence: when control variables are added to the model, unobserved factors are no longer related to variable of interest

  • Mathematically, assumption is \(E[u_{i}|X_{1i}, X_{2i}] = E[u_{i}|X_{2i}]\)

    • When you control for \(X_{2i}\), average of unobserved factors do not depend on \(X_{1i}\)

  • Including \(X_{2i}\) breaks the correlation between \(X_{1i}\) and \(u_{i}\)

    • We allow \(X_{2i}\) to be correlated with error term

    • Through its correlation with the error, \(X_{2i}\) holds constant itself and any related unobserved factor

    • Estimate of \(\beta_{1}\) is unbiased if no other relevant unobserved factors are related to \(X_{1i}\)

Control Variables in Multiple Regression

  • Estimate of \(\beta_{2}\) is not necessarily unbiased

    • We allow \(X_{2i}\) to be correlated with error term

      • When this correlation exists, its slope estimate is biased

    • This is okay, because it is not the variable of interest

    • Key is that estimate of \(\beta_{1}\) is unbiased

  • Ex: Effect of class size on test scores \[TestScore_{i} = \beta_{0} + \beta_{1}STR_{i} + \beta_{2}PctEL_{i} + \beta_{3}LchPct_{i} + u_{i}\]

  • Variable of interest is \(STR_{i}\)

  • We are worried that \(STR_{i}\) is related to unobserved factors

    • e.g., unobserved student economic background

    • Without controlling for economic background, we have omitted variables bias in \(\hat{\beta}_{1}\)

Control Variables in Multiple Regression

  • To control for economic background, include \(PctEL_{i}\) and \(LchPct_{i}\) as control variables

    • Holds constant direct effects of \(PctEL_{i}\) and \(LchPct_{i}\)

    • Also holds constant other related economic background factors

  • If conditional independence assumption holds, \(\hat{\beta}_{1}\) is unbiased

    • With \(PctEL_{i}\) and \(LchPct_{i}\) in model, \(STR_{i}\) is not related to unobserved factors

  • Note that \(\hat{\beta}_{2}\) and \(\hat{\beta}_{3}\) are in general biased

    • \(PctEL_{i}\) and \(LchPct_{i}\) are related to relevant unobserved factors

      • They control for those unobserved factors

      • This is what makes them good control variables

    • Relationship with unobserved factors means their slope estimates are biased

Conditional Mean Independence

  • Suppose regression model is \[Y_{i} = \beta_{0} + \beta_{1}X_{1i} + \beta_{2}X_{2i} + u_{i}\]

  • For this model, imagine \(E[u_{i}|X_{1i}, X_{2i}] = 0\) is not true

    • Regressors might be correlated with unobserved factors

  • But we can assume \(E[u_{i}|X_{1i}, X_{2i}] = E[u_{i}|X_{2i}]\)

    • With \(X_{2i}\) in the model, \(X_{1i}\) is not related to unobserved factors

      • But, \(X_{2i}\) can be related to them

  • If second assumption is true, we can get unbiased estimate of \(\beta_{1}\), but not \(\beta_{2}\)

Conditional Mean Independence

  • Imagine the error is linearly related to \(X_{1i}\) and \(X_{2i}\) \[u_{i} = \gamma_{0} +\gamma_{1}X_{1i}+ \gamma_{2}X_{2i} + v_{i}\]

    • Assume error in this equation \(v_{i}\) is unrelated to \(X_{1i}\) and \(X_{2i}\)

  • Conditional mean independence is equivalent to \(\gamma_{1}=0\)

  • The error equation under this assumption is \[u_{i} = \gamma_{0} + \gamma_{2}X_{2i} + v_{i}\]

  • From this you can see \(E[u_{i}| X_{1i},X_{2i}] = E[u_{i}|X_{2i}]\)

    • The expected value of \(u_{i}\) given \(X_{1i}\) and \(X_{2i}\) is \[E[u_{i}| X_{1i},X_{2i}]= \gamma_{0} + \gamma_{2}X_{2i}\]

    • Because \(X_{1i}\) is not in the equation, this is the same as \[E[u_{i}|X_{2i}]= \gamma_{0} + \gamma_{2}X_{2i}\]

Conditional Mean Independence

  • Substitute \(u_{i}\) equation into original regression \[Y_{i} = \beta_{0} + \beta_{1}X_{1i} + \beta_{2}X_{2i} +\gamma_{0} + \gamma_{2}X_{2i} + v_{i}\]

  • Collecting terms, \[Y_{i} = (\beta_{0}+ \gamma_{0}) + \beta_{1}X_{1i} + (\beta_{2} + \gamma_{2})X_{2i} + v_{i}\]

  • Equivalently, \[Y_{i} = \delta_{0} + \beta_{1}X_{1i} + \delta_{2} X_{2i} + v_{i}\]

  • This model satisfies the 4 OLS assumptions

  • So if we estimate a regression of \(Y_{i}\) on \(X_{1i}\) and \(X_{2i}\)

    • We get unbiased estimate of \(\beta_{1}\)

    • And unbiased estimate of \(\delta_{2}\)

      • Because \(\delta_{2}=\beta_{2} + \gamma_{2}\), this is not unbiased for \(\beta_{2}\)

Model Specification in Practice

  • Main reason to add variables to a regression is to control omitted variables bias

  • Requires knowing what factors determine \(Y\) that relate to \(X\)

    • Obtaining background knowledge on the economic question is crucial

  • Even with some background knowledge, judgement calls are necessary

    • Experts often disagree on appropriate control variables

    • And in the end, conditional independence is an unverifiable assumption

Model Specification in Practice

  • In practice, researchers present results from several model specifications

    • Base specification: basic regression containing variables of interest and control variables suggested by theory and judgement

    • Alternative specification: regressions with alternative sets of regressors, often the ones in base specification plus more

  • If slope estimate on variable of interest is unchanged across specifications, that is evidence that it is unbiased

    • Big changes in main estimate would be signal of omitted variables bias

  • However, it is not indisputable proof of unbiased estimates

    • There could still be omitted variables

Interpreting \(R^2\) and \(\bar{R}^2\) in Practice

  • \(R^2\) and \(\bar{R}^2\) are not useful for deciding to add/drop variables

  • There are several pitfalls when using these measures

  1. Increases in \(R^2\) and \(\bar{R}^2\) do not mean a variable is significant

    • \(R^2\) always increases with an additional regressor

    • \(\bar{R}^2\) sometimes increases when regressor overcomes “penalty”

    • Neither mean new variable is statistically significant

  2. A high \(R^2\) or \(\bar{R}^2\) does not mean regressors cause changes in dependent variable

    • \(R^2\) and \(\bar{R}^2\) measure only how model fits the sample

    • Causality is related to relationship between regressors and error

    • These are independent concepts

Interpreting \(R^2\) and \(\bar{R}^2\) in Practice

  1. A high \(R^2\) or \(\bar{R}^2\) does not mean there is no omitted variables bias

    • Again, \(R^2\) and \(\bar{R}^2\) measure only how model fits the sample

    • Omitted variables bias is related to relationship between error and X

    • These concepts are not directly related

  2. A high \(R^2\) or \(\bar{R}^2\) does not mean you have the most appropriate set of regressors

    • The “right” set of regressors is a difficult concept

    • There are varying opinions on what should be in a model

    • Partly it is driven by fit, partly by data availability, partly by theory

Analysis of Test Score Data

Introduction

  • Here we demonstrate the model specification lessons with an example

  • As in the past, we examine the relationship between class size and test scores

  • Also learn some practical advice for reporting regression results

    • Includes discussion on scale of the variables in model

    • Also how researchers typically structure output in a table

    • And how to present graphs

Base and Alternative Specifications

  • We discussed base and alternative specifications

    • Base contains variables of primary interest

    • Alternative specifications test different sets of regressors

  • Before showing results, need to identify variable of interest

  • We are interested in effect of class size on test scores

    • So variable of interest is class size

  • Control variables added to hold constant relevant factors related to class size

    • Many factors affect test scores and are related to class size

      • Outside learning opportunities

      • Economic background

Base and Alternative Specifications

  • As control variables, we have the following

    • Percent of students ESL

    • Percent eligible for free/subsidized lunch

    • Percent who qualify for income assistance

  • These variables control for their direct effects and related unobserved factors

    • Economic background

    • Measures of economic disadvantage

  • Estimate of class size effect unbiased when conditional independence is true

    • If adding variables above breaks link between error and class size, conditional independence is satisfied

    • Control variables hold constant omitted factors

Base and Alternative Specifications

  • Base specification will be regression of test score on student-teacher ratio

  • Alternatives will add different sets of control variables

  • We examine how estimate on student teacher ratio changes

    • Big changes suggest there was omitted variables bias

    • No changes suggest the opposite

Scale of the Regressors

  • All regressors are measured on a particular scale

    • In example all are measured in percentage points

  • In practice you can change the scale for easier interpretation

Scale of the Regressors

  • Example: percentage points vs fraction

    • Percentage points range from 0 to 100

    • Decimal fraction measures same thing from 0 to 1

    • Decimal fraction equals percentage divided by 100

      • Ex: if half of people get free/reduced lunch, then percentage is 50 and fraction is 0.5

  • Changing scale alters interpretation of regression slope

  • Slope always measures effect of 1-unit change in \(X\) on \(Y\)

    • If free lunch measured as percent, one unit is 1 percentage point

    • If free lunch measured as fraction, one unit is 100 percentage points

      • Moving fraction from 0 to 1 is a 100 percentage point change

Scale of the Regressors

  • Common to change scale for \(X\) variables like income

    • If measured in dollars, slope measures effect of $1 change

      • Often produces very small coefficients

      • $1 change is usually not policy relevant

    • If you divide income by $1000, it is measured in thousands of dollars

    • In this case slope measures effect of $1000 change

      • Much easier to interpret

      • More policy relevant

  • Choosing scale of variables is where statistics meets arts

    • A matter of personal preference which scale to use

Graphical Presentation of Data

Graphical Presentation of Data

  • Previous slide shows three scatterplots

    • Test scores against percent ESL

    • Test scores against percent free/reduced lunch

    • Test scores against percent income assistance

  • Researchers present graphs like these as descriptive evidence

    • Shows correlation of each variable with test scores

  • Based on this, we expect all variables are negatively related to test scores

    • Clearest relationship is with free/reduced lunch

  • These graphs are purely descriptive

    • Do not show causal relationships, only correlation

    • Nothing is held constant when producing these plots

Tabular Presentation of Results

Tabular Presentation of Results

  • Previous slide is typical of how regression results are presented

  • Results from several specifications grouped into one table

    • First is base specification

    • Other four are alternative specifications

  • To evaluate omitted variables bias, look at estimate on student-teacher ratio across columns

    • Big drop in estimate when control variables added

    • Then stays relatively stable with different sets of controls

    • Suggests that controls absorb effect of some omitted factors

  • Measures of fit presented at bottom

    • Can compare these across regressions to see how they change

    • Fit improves with more variables

Tabular Presentation of Results

  • Other notable features of this table

    • Standard errors are in brackets under estimate

    • Stars indicate statistical significance

      • * is significant at 5% level

      • ** is significant at 1% level

    • Footnote at bottom describes important details

      • Heteroskedasticity-robust standard errors used

      • Short description of data

  • Main point of table is to present key information in easy to read format

Discussion of Results

  • Controlling for student background matters

    • Effect of class size on test scores falls by half

  • Class size is statistically significant at 5% level in all regressions

    • We reject null that it is zero

  • The set of control variables does not matter much

    • A slight difference in class size effect when we measure background with free/reduced lunch vs income assistance

  • The model fits the data well

    • Adjusted \(R^2\) high when controls are added

Example with Stata

  • Here we repeat the above with the birthweight example

  • Key variable of interest is packs

    • Want unbiased estimate of effect of smoking on birthweight

  • Control variables are education and income

    • Controls for their direct effects

    • And other factors related, like socio-economic status and health care use

  • First examine effect of scaling education variable

  • Then present base and alternative specifications

Example with Stata

regress bwghtlbs packs motheduc fatheduc faminc, robust
Linear regression                               Number of obs     =      1,191
                                                F(4, 1186)        =      11.47
                                                Prob > F          =     0.0000
                                                R-squared         =     0.0328
                                                Root MSE          =     1.2401

------------------------------------------------------------------------------
             |               Robust
    bwghtlbs | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       packs |  -.7368692   .1303338    -5.65   0.000    -.9925797   -.4811587
    motheduc |  -.0273702   .0184606    -1.48   0.138    -.0635892    .0088488
    fatheduc |   .0308543   .0165627     1.86   0.063     -.001641    .0633497
      faminc |   .0033641    .002215     1.52   0.129    -.0009817    .0077099
       _cons |   7.379629   .1995718    36.98   0.000     6.988075    7.771182
------------------------------------------------------------------------------

Example with Stata

gen motheduc2 = motheduc/10
regress bwghtlbs packs motheduc2 fatheduc faminc, robust
Linear regression                               Number of obs     =      1,191
                                                F(4, 1186)        =      11.47
                                                Prob > F          =     0.0000
                                                R-squared         =     0.0328
                                                Root MSE          =     1.2401

------------------------------------------------------------------------------
             |               Robust
    bwghtlbs | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       packs |  -.7368692   .1303338    -5.65   0.000    -.9925797   -.4811587
   motheduc2 |  -.2737022   .1846057    -1.48   0.138    -.6358924     .088488
    fatheduc |   .0308543   .0165627     1.86   0.063     -.001641    .0633497
      faminc |   .0033641    .002215     1.52   0.129    -.0009817    .0077099
       _cons |   7.379629   .1995718    36.98   0.000     6.988075    7.771182
------------------------------------------------------------------------------

Example with Stata

qui regress bwghtlbs packs, robust
qui estimates store base
qui regress bwghtlbs packs motheduc fatheduc , robust
qui estimates store alt1
qui regress bwghtlbs packs motheduc fatheduc faminc, robust
qui estimates store alt2
            
estout base alt1 alt2, cells(b(star fmt(3)) se(par fmt(3))) starlevels(* 0.05 ** 0.01 )
                     base           alt1           alt2  
                     b/se           b/se           b/se  
---------------------------------------------------------
packs              -0.775**       -0.749**       -0.737**
                  (0.128)        (0.131)        (0.130)  
motheduc                          -0.022         -0.027  
                                 (0.018)        (0.018)  
fatheduc                           0.037*         0.031  
                                 (0.016)        (0.017)  
faminc                                            0.003  
                                                (0.002)  
_cons               7.539**        7.330**        7.380**
                  (0.038)        (0.198)        (0.200)  
---------------------------------------------------------