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## 4.6 Exercises

#### 1. Class Sizes and Test Scores

A researcher wants to analyze the relationship between class size (measured by the student-teacher ratio) and the average test score. Therefore he measures both variables in \(10\) different classes and ends up with the following results.

Class Size |
23 | 19 | 30 | 22 | 23 | 29 | 35 | 36 | 33 | 25 |

Test Score |
430 | 430 | 333 | 410 | 390 | 377 | 325 | 310 | 328 | 375 |

**Instructions:**

Create the vectors

`cs`(the class size) and`ts`(the test score), containing the observations above.Draw a scatterplot of the results using

`plot()`.

#### 2. Mean, Variance, Covariance and Correlation

The vectors `cs` and `ts` are available in the working environment (you can check this: type their names into the console and press enter).

**Instructions:**

Compute the mean, the sample variance and the sample standard deviation of

`ts`.Compute the covariance and the correlation coefficient for

`ts`and`cs`.

**Hint:** Use the `R` functions presented in this chapter: `mean()`, `sd()`, `cov()`, `cor()` and `var()`.

#### 3. Simple Linear Regression

The vectors `cs` and `ts` are available in the working environment.

**Instructions:**

The function

`lm()`is part of the package`AER`. Attach the package using`library()`.Use

`lm()`to estimate the regression model \[TestScore_i = \beta_0 + \beta_1 STR_i + u_i.\] Assign the result to`mod`.Obtain a statistical summary of the model.

#### 4. The Model Object

Let us see how an object of class `lm` is structured.

The vectors `cs` and `ts` as well as the model object `mod` from the previous exercise are available in your workspace.

**Instructions:**

- Use
`class()`to learn about the class of the object`mod`. `mod`is an object of type`list`with named entries. Check this using the function`is.list()`.- See what information you can obtain from
`mod`using`names()`. - Read out an arbitrary entry of the object
`mod`using the`$`operator.

#### 5. Plotting the Regression Line

You are provided with the code for the scatterplot in `script.R`

**Instructions:**

Add the regression line to the scatterplot from a few exercises before.

The object

`mod`is available in your working environment.

**Hint:** Use the function `abline()`.

#### 6. Summary of a Model Object

Now read out and store some of the information that is contained in the output of `summary()`.

**Instructions:**

Assign the output of

`summary(mod)`to the variable`s`.Check entry names of the object

`s`.Create a new variable

`R2`and assign the \(R^2\) of the regression.

The object `mod` is available in your working environment.

#### 7. Estimated Coefficients

The function `summary()` also provides information on the statistical significance of the estimated coefficients.

**Instructions:**

Extract the named \(2\times4\) matrix with estimated coefficients, standard errors, \(t\)-statistics and corresponding \(p\)-values from the model summary `s`. Save this matrix in an object named `coefs`.

The objects `mod` and `s` are available in your working environment.

#### 8. Dropping the Intercept

So far, we have estimated regression models consisting of an intercept and a single regressor. In this exercise you will learn how to specify and how to estimate regression a model without intercept.

Note that excluding the intercept from a regression model might be a dodgy practice in some applications as this imposes the conditional expectation function of the dependent variable to be zero if the regressor is zero.

**Instructions:**

Figure out how the

`formula`argument must be specified for a regression of`ts`solely on`cs`, i.e., a regression without intercept. Google is your friend!Estimate the regression model without intercept and store the result in

`mod_ni`.

The vectors `cs`, `ts` and the model object `mod` from previous exercises are available in the working environment.

#### 9. Regression Output: No Constant Case

In Exercise 8 you have estimated a model without intercept. The estimated regression function is

\[\widehat{TestScore} = \underset{(1.36)}{12.65} \times STR.\]

**Instructions:**

Convince yourself that everything is as stated above: extract the coefficient matrix from the summary of `mod_ni` and store it in a variable named `coef`.

The vectors `cs`, `ts` as well as the model object `mod_ni` from the previous exercise are available in your working environment.

**Hint:** An entry of a named list can be accessed using the `$` operator.

#### 10. Regression Output: No Constant Case — Ctd.

In Exercises 8 and 9 you have dealt with a model without intercept. The estimated regression function was

\[\widehat{TestScore_i} = \underset{(1.36)}{12.65} \times STR_i.\]

The coefficient matrix `coef` from Exercise 9 contains the estimated coefficient on \(STR\), its standard error, the \(t\)-statistic of the significance test and the corresponding \(p\)-value.

**Instructions:**

- Print the contents of
`coef`to the console. - Convince yourself that the reported \(t\)-statistic is correct: use the entries of
`coef`to compute the \(t\)-statistic and save it to`t_stat`.

The matrix `coef` from the previous exercise is available in your working environment.

**Hints:**

`X[a,b]`returns the`[a,b]`element of the matrix`X`.The \(t\)-statistic for a test of the hypothesis \(H_0: \beta_1 = 0\) is computed as \[t = \frac{\hat{\beta}_1}{SE(\hat{\beta}_1)}.\]

#### 11. Two Regressions, One Plot

The two estimated regression models from the previous exercises are

\[\widehat{TestScore_i} = \underset{(1.36)}{12.65} \times STR_i\]

and

\[\widehat{TestScore_i} = \underset{(23.96)}{567.4272} \underset{(0.85)}{-7.1501} \times STR_i.\]

You are provided with the code line `plot(cs, ts)` which creates a scatterplot of `ts` and `cs`. Note that this line must be executed before calling `abline()`! You may color the regression lines by using, e.g., `col = “red”` or `col = “blue”` as an additional argument to `abline()` for better distinguishability.

The vectors `cs` and `ts` as well as the list objects `mod` and `mod_ni` from previous exercises are availabe in your working environment.

**Instructions:**

Generate a scatterplot of `ts` and `cs` and add the estimated regression lines of `mod` and `mod_ni`.

#### 12. \(TSS\) and \(SSR\)

If graphical inspection does not help, researchers resort to analytic techniques in order to detect if a model fits the data at hand good or better than another model.

Let us go back to the simple regression model including an intercept. The estimated regression line for `mod` was

\[\widehat{TestScore_i} = 567.43 - 7.15 \times STR_i, \, R^2 = 0.8976, \, SER=15.19.\]

You can check this as `mod` and the vectors `cs` and `ts` are available in your working environment.

**Instructions:**

- Compute \(SSR\), the sum of squared residuals, and save it to
`ssr`. - Compute \(TSS\), the total sum of squares, and save it to
`tss`.

#### 13. The \(R^2\) of a Regression Model

The \(R^2\) of the regression saved in `mod` is \(0.8976\). You can check this by executing `summary(mod)$r.squared` in the console below.

Remember the formula of \(R^2\):

\[R^2 = \frac{ESS}{TSS} = 1 - \frac{SSR}{TSS}\]

The objects `mod`, `tss` and `ssr` from the previous exercise are available in your working environment.

**Instructions:**

- Use
`ssr`and`tss`to compute \(R^2\) manually.*Round*the result to*four*decimal places and save it to`R2`. - Use the logical operator
`==`to check whether your result matches the value mentioned above.

**Hints:**

You may round numeric values using the function `round()`.

#### 14. The Standard Error of The Regression

The standard error of the Regression in the simple regression model is \[SER = \frac{1}{n-2} \sum_{i=1}^n \widehat{u}_i^2 =\sqrt{\frac{SSR}{n-2}}.\] \(SER\) measures the size of an average residual which is an estimate of the magnitude of a typical regression error.

The model object `mod` and the vectors `cs` and ts are available in your workspace.

**Instructions:**

Use

`summary()`to obtain the \(SER\) for the regression of`ts`on`cs`saved in the model object`mod`. Save the result in the variable`SER`.Use

`SER`to compute the \(SSR\) and store it in`SSR`.Check that

`SSR`is indeed the \(SSR\) by comparing`SSR`to the result of`sum(mod$residuals^2)`

#### 15. The Estimated Covariance Matrix

As has been discussed in Chapter 4.4, the OLS estimators \(\widehat{\beta}_0\) and \(\widehat{\beta}_1\) are functions of the random error term. Therefore, they are random variables themselves. For two or more random variables, their covariances and variances are summarized by a *variance-covariance matrix* (which is often simply called the *covariance matrix*). Taking the square root of the diagonal elements of the estimated covariance matrix obtains \(SE(\widehat\beta_0)\) and \(SE(\widehat\beta_1)\), the standard errors of \(\widehat{\beta}_0\) and \(\widehat{\beta}_1\).

`summary()` computes an estimate of this matrix. The respective entry in the output of summary (remember that `summary()` produces a list) is called `cov.unscaled`. The model object `mod` is available in your workspace.

**Instructions:**

Use

`summary()`to obtain the covariance matrix estimate for the regression of test scores on student-teacher ratios stored in the model object`mod`. Save the result to`cov_matrix`.Obtain the diagonal elements of

`cov_matrix`, compute their square root and assign the result to the variable`SEs`.

**Hint:** `diag(A)` returns a vector containing the diagonal elements of the matrix `A`.