Further Exercises for MW24.1 - 09

1. Question

   Install R and a frontend for R, e.g. R-Studio.

   Save the file DYQ.csv in your working directory. (Don’t take the detour of opening the file in your spreadsheet program (e.g. Microsoft Excel) and save it from that program. Spreadsheet programs often do considerable damage to your data. By taking the detour through your spreadsheet program you will often change numbers to what the spreadsheet program thinks is a date. If you find it difficult to save a link from your browser, try clicking with the right mouse button on the link. In many browsers this right-click opens a menu where you can choose “Save link as…” or something similar.)

   If you are not sure which directory R currently uses as a working directory, you can use the command

   getwd()

   In RStudio you can also use the menu Session / Set Working Directory / Choose Directory to choose your working directory.

   Read the file DYQ.csv with the command

   DYQ <- read.csv("DYQ.csv")

   Now the variable DYQ contains your data. In this exercise we have a brief look at your data:

   
   
   1. Use the command nrow(DYQ) to determine the number of rows in your data. How many rows do you have?
   2. Use the command names(DYQ) to determine the names of the variables in your data. How many variables do you have?
   3. Use the command mean(DYQ$X3) to determine the mean of the variable X3 in your data.
   4. Use the command median(DYQ$X3) to determine the median of the variable X3 in your data.
   5. Use the command sd(DYQ$X3) to determine the standard deviation of the variable X3 in your data. (If Moodle complains about an “incomplete answer”, please check whether the format of your answer is in line with Moodle’s expectations. Make sure that Moodle and you interpret decimal separators in the same way. Depending on the settings of your computer it is possible that Moodle expects decimal numbers like 3.14 and not like 3,14).
   
   

2. Question

   Your sample of the random variable $X$ contains $n=9$ independent and normally distributed observations: $X_1,\ldots,X_{9}$. You are looking for an estimator for $E[X]$. Which of the following statements are correct:

   1. The estimator $-\frac{1}{3}X_{2}-\frac{1}{3}X_{3}+\frac{1}{3}X_{5}+\frac{4}{3}X_{7}$ is an unbiased estimator for $E[X]$.

      Yes / No

   2. The estimator $-2X_{1}+2X_{3}-\frac{1}{2}X_{5}+\frac{3}{2}X_{6}$ is an unbiased estimator for $E[X]$.

      Yes / No

   3. The estimator $\frac{1}{2}X_{5}-X_{6}-2X_{8}-2X_{9}$ is an unbiased estimator for $E[X]$.

      Yes / No

   4. The estimator $2X_{3}-X_{4}-X_{7}$ is an unbiased estimator for $E[X]$.

      Yes / No

   5. The estimator $X_{4}$ dominates $\frac{1}{6}X_{1}+\frac{2}{3}X_{2}+\frac{2}{3}X_{4}-\frac{1}{2}X_{6}$.

      Yes / No

   6. The estimator $\frac{1}{5}X_{1}+\frac{3}{10}X_{2}+\frac{1}{10}X_{6}+\frac{2}{5}X_{9}$ dominates $-\frac{1}{2}X_{2}+X_{3}+\frac{1}{2}X_{5}$.

      Yes / No

   
   
   
   

3. Question

   You use a level of significance of 0.01.

   
   
   1. You assume that your test statistic follows a standard normal distribution. How large (in absolute terms) can your test statistic (for a two-sided test) be, so that you still don’t reject your Null-hypothesis? (You can calcualate this value with R.)
   2. You assume that the random variable $X$ follows a normal distribution with unknown mean and standard deviation 6. Your sample contains 32 observations. The sample mean is -5. Your Null-hypothesis is that $X$ has a mean of 20. How large is the absolute value of your test statistic?
   3. You still assume that the random variable $X$ follows a normal distribution with unknown mean and standard deviation 6. Now you consider a sample with 32 observations and sample mean 0.7. Your Null-hypothesis is still that $X$ has a mean of 20. How large is the $p$-value (for a two-sided test, rounded to 4 decimal places)?
   
   

4. Question

   Your data in the file DHC.csv contains two variables: X and f. The variable f tells you which group (C or D) the observation X belongs to.

   Compare the mean of X for two groups C and D with the help of a (two-sided) $t$-test.

   1. Your Null-hypothesis is that the mean of the normally distributed X is the same in both groups. How large is the $p$-value for this $t$-test?
   2. You use a level of significance of 5%. Do you reject your Null-hypothesis? Yes / No
   
   
   
   

5. Question

   The random variable $X \sim N(\mu,\sigma^2)$ follows a normal distribution with unknown variance $\sigma^2$. You draw a sample with 24 observations. You find a sample mean of 7 and a sample standard deviation of 9.

   • Determine a 95%-confindence interval for your estimation of the expected value of $X$: $\hat{\mu}$.
   
   
   1. What is the lower boundary of the interval?
   2. What is the upper boundary of the interval?
   
   

6. Question

   A random variable $X$ is distributed as follows:

   • $P(X=A)=\theta$
   • $P(X=B)=2\theta$
   • $P(X=C)=1-3\theta$

   We have $\theta\in[0,1/3]$.

   In your sample you have the following observations:

   $\{ B, C, B, A, B \}$.

   What is the maximum-likelihood estimator for $\theta$?

   
   

7. Question

   The file DCF.csv contains two variables: X and Y.

   To explain Y as a linear function of X, you estimate the model

   $Y = \beta_0 + \beta_1 X + u$.

   1. Which value do you estimate for $\beta_1$?
   2. Your (two sided) Null-hypothesis is $H_0: \beta_1=0$. Determine the $p$-value for this test (report at least 4 decimal places).
   3. You use a level of significance of 10%. Can you reject your Null-hypothesis? Yes / No
   4. What is the lower boundary of the 95% confidence interval for $\beta_1$?
   5. What is the upper boundary of the 95% confidence interval for $\beta_1$?
   
   
   
   

8. Question

   Use data from the file D03b.csv.

   You want to measure the effect X1 has on Y1, the effect X2 has on Y2, the effect X3 has on Y3, and the effect X4 has on Y4. For each case below, select the most suitable specification and provide the point estimate of the effect.

   
   
   1. By how many percentage points does Y1 change approximately when X1 changes by one unit?
   2. What is the elasticity of Y2 with respect to X2?
   3. What is the marginal effect of X3 on Y3?
   4. By which amount does Y4 change when X4 changes by 1 percentage point?
   
   

9. Question

   Use the data from the file DUX.csv. Based on this data you estimate the following relationship:

   $Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + u$.

   1. Which value do you estimate for $\beta_2$?
   2. Your (two sided) Null-hypothesis is $H_0: \beta_2 = 0$. Determine the $p$-value for this test (report at least 4 decimal places).
   3. You use a level of significance of 1%. Can you reject your Null-hypothesis? Yes / No
   4. What is the lower boundary of the 95%-confidence-interval for $\beta_2$?
   5. What is the upper boundary of the 95%-confidence-interval for $\beta_2$?
   
   
   
   

10. Question

    Consider the following model:

    $tv = \beta_0 + \beta_1 \cdot ba + \beta_2 \cdot kh + \beta_3 \cdot ba \cdot kh + u$

    The variable $ba$ indicates whether you are in situation JMF or VSG: In the case of JMF you have $ba=0$. In the case of VSG you have $ba=1$.

    The variable $kh$ indicates whether you are in situation WEH or YCA: In the case of WEH you have $kh=0$. In the case of YCA you have $kh=1$.

    The mean values of $tv$ for the four different combinations of JMF and VSG and WEH and YCA are shown in the following table:

    

    
    
    1. What is $\beta_0$?
    2. What is $\beta_1$?
    3. What is $\beta_2$?
    4. What is $\beta_3$?
    
    

11. Question

    Save the file DLE.csv in your working directory. You want to estimate the mean of the absolute value of X1.

    
    
    1. What is the plug-in estimate of the mean of the absolute value of X1?
    2. Use a bootstrap (with 10000 replications) to determine the standard deviation of this estimate
    3. You assume this estimate follows a normal distribution. Use a parametric bootstrap to determine the lower boundary of a 95%-confidence interval for the mean of the absolute value of X1?
    
    

12. Question

    The random variable $T$ follows a normal distribution with unknown mean $\mu_{T}$ and known standard deviation $\sigma_{T}=7$.

    According to your prior the following holds:

    • $\mu_{T}=6$ with probability 2/7,
    • $\mu_{T}=14$ with probability 5/7.

    The probability for all other values of $\mu_{T}$ is zero.

    You have one observation, $T=4$.

    In the following you can use dnorm to calculate the density function of the normal distribution.

    E.g. dnorm(4,6,7) yields the density of the normal distribution for $T=4$ when $\mu_{T}=6$ and $\sigma_{T}=7$.

    
    
    1. What is the posterior probability $P(\mu_{T}=6| T=4 )$?
    2. What is the posterior probability $P(\mu_{T}=10| T=4 )$?
    3. Now you have two observations: $T=\{4,9\}$. What is the posterior probability $P(\mu_{T}=6| T=\{ 4,9 \} )$?
    
    

13. Question

    The file DWW.csv contains a variable X. This X is a sample of the random variable $X$. Here we write the normal distribution as $N(\mu,\tau)$ where $\mu$ is the mean and $\tau=1/\sigma^2$ is the precision. You assume that $X$ follows a normal distribution: $X \sim N(\mu,1/\sigma^2)$ where $\sigma^2$ is the variance of $X$. You have the following priors: $\mu \sim N(0,.0001)$, $\tau=1/\sigma^2\sim \Gamma(.01,.01)$ ($\Gamma$ denotes the Gamma distribution).

    To obtain the necessary precision, please use run.jags defaults. Please don’t change options or modules.

    
    
    1. What is the lower boundary of the 95%-credible-interval for $\mu$?
    2. What is the upper boundary of the 95%-credible-interval for $\mu$?
    3. What is the lower boundary of the 95%-credible-interval for $\sigma$?
    4. What is the upper boundary of the 95%-credible-interval for $\sigma$?
    5. What is the posterior probability (a number between 0 and 1) of $\mu>7.43$?
    6. What is the posterior probability (a number between 0 and 1) of $3.69<\mu<5.64$?
    
    

14. Question

    The file DTN.csv contains an independent variable F and a dependent binary variable L.

    You estimate the following model:

    $$P(L=1|F=f) = F(\beta_0 + \beta_F f)$$

    where $F$ is the logistic distribution.

    
    
    1. What is your estimate for $\beta_F$?
    2. What is the marginal effect of $f$ for the average value of $F$ in your data?
    3. What is the average marginal effect of $f$?
    4. What is the marginal effect of $f$ if $f=1.738$?
    5. What are the odds for $L=1$ if $f=1.738$?
    
    

15. Question

    The file DVU.csv contains two independent variables X1, and X2, and a dependent count variable Y. (Hint: In the following you may find the library MASS useful.)

    
    
    1. Use a Poisson model where you explain Y as a function of X1 and X2. What is the coefficient of X2?
    2. Now you use a negative binomial model to explain Y as a function of X1 and X2. What is now the coefficient of X2?
    3. In the negative binomial model, what is your estimate for the parameter $\theta$?
    4. Your Null-hypothesis is that $\theta=\infty$, i.e. that the negative binomial model does not significantly improve the goodness of fit of the Poisson model. Use a Likelihood-Ratio test to test this hypothesis. Which $p$-value do you get (rounded to 4 decimal places)?
    
    

16. Question

    The file DAR.csv contains three variables, a, H and X. The variable a denotes to which group observations belong. X is our dependent variable.

    We write the normal distribution as $N(\mu,\tau)$ where $\mu$ is the mean and $\tau=1/\sigma^2$ is the precision. $\Gamma$ denotes the Gamma distribution. You use JAGS to estimate the following model with random effects:

    $$X_{at} = \beta_0 + \nu_{a} + \epsilon_{at}$$

    where the group specific random effect $\nu_a \sim N(0,\tau_\nu)$ and the residual $\epsilon_{at} \sim N(0,\tau_\epsilon)$. Here $\tau_\nu=1/\sigma^2_\nu$ and $\tau_\epsilon=1/\sigma^2_\epsilon$ are the precision of $\nu_a$ and $\epsilon_{at}$, respectively.

    Your priors are: $\beta_0 \sim N(0,0.0001)$, $\tau_\nu \sim \Gamma(.01,.01)$, $\tau_\epsilon \sim \Gamma(.01,.01)$.

    To obtain the necessary precision, please use run.jags defaults. Please don’t change the options or modules.

    
    
    1. What is the 50%-quantile of your posterior for $\sigma_\epsilon$?
    2. What is the 50%-quantile of your posterior for $\sigma_\nu$?
    3. What is the posterior probability (a number between 0 and 1) of $\sigma_\nu>3.3$?
    
    

17. Question

    The file DBN.csv contains the variables X, Y and Z. You estimate the model $Y = \beta_0 + \beta_1 X + u$.

    
    
    1. Use a standard OLS model to estimate the model. What is the coefficient of $X$?
    2. Provide a $p$-value for the test of the Null-hypothesis that the coefficient of $X$ is zero (round to 4 decimal places)?
    3. Now use the variable $Z$ as an instrument for $X$. Use the command ivreg from the AER library to estimate the coefficient of $X$ for this model.
    4. For the instrumental variables model provide a $p$-value for the test of the Null-hypothesis that the coefficient of $X$ is zero (round to 4 decimal places)?
    
    

18. Question

    The file DJX.csv contains eight independent variables, X1, X2, X3, X4, X5, X6, X7, X8, and a dependent variable Y. You estimate the following (full) model:

    $Y = \beta_0 + \beta_{1} X_{1} + \beta_{2} X_{2} + \beta_{3} X_{3} + \beta_{4} X_{4} + \beta_{5} X_{5} + \beta_{6} X_{6} + \beta_{7} X_{7} + \beta_{8} X_{8} + u$

    
    
    1. What is the coefficient of X2 in the full model?
    2. You simplify the model and include only terms which are significant on a 5% level in the above estimation. You drop insignificant terms only once. If you find insignificant terms in your simplified model, you keep them. You also keep X2. What is now the coefficient of X2?
    3. Use the function extractAIC to obtain the AIC of this (simplified) model. (Note: the function extractAIC returns two numbers. Only one of them is the AIC).
    4. Now you use the step function to simplify the (full) model based on the AIC. If the step function has removed X2 from the model, add X2 back to your model. What is the coefficient of X2 in this model?
    5. Use the function extractAIC to obtain the AIC of this model.
    
    

19. Question

    The file DYW.csv contains an independent variable H and a dependent binary variable V.

    You estimate the following model:

    $$P(V=1|[H=h) = \Phi(\beta_0 + \beta_{h} h)$$

    where $\Phi$ is the standard normal distribution.

    
    
    1. What is your estimate for $\beta_{h}$?
    2. What is the marginal effect of $h$ for the average value of $h$ in your data?
    3. What is the average marginal effect of $h$?
    4. What is the marginal effect of $h$ if $h=-0.348$?
    
    

20. Question

    A random variable $X$ follows a distribution with density function $f(x|\theta)=\left(\frac {x}{23}\right)^\theta \cdot \frac{\theta}{x}$ if $x\in[0,23]$ and $f(x)=0$ otherwise.

    Your sample contains the observations $\{ 3, 14, 14, 18 \}$.

    What is the Maximum-Likelihood estimator for $\theta$?

    
    

21. Question

    The data in the file DGX.csv contains 8 variables, Ybz, Yjr, Ykv, Ysf, Xbz, Xjr, Xkv, Xsf. You investigate the effect Xbz has on Ybz, the effect Xjr has on Yjr, the effect Xkv has on Ykv, the effect Xsf has on Ysf. For each case below, select the most suitable specification and provide the point estimate of the effect.

    1. Use a specification where Ybz changes by a fixed number of percentage points when Xbz changes by one unit. By how many percentage points does Ybz change approximately when Xbz changes by one unit?
    2. Use a specification where Ysf changes by a fixed amount when Xsf changes by a given percentage. By which amount does Ysf change when Xsf changes by 1 percentage point?
    3. Use a specification where the marginal effect of Xkv on Ykv is constant. What is the marginal effect of Xkv on Ykv?
    4. Use a specification where the elasticity of Yjr with respect to Xjr is constant. What is the elasticity of Yjr with respect to Xjr?
    
    
    
    

22. Question

    The file DTR.csv contains a variable A. This A is a sample of the random variable $A$. You assume that $A$ follows a normal distribution: $A \sim N(\mu,1/\sigma^2)$ where $\sigma^2$ is the variance of $A$. Your priors are $\mu \sim N(-10.6,8)$, $\tau=1/\sigma^2\sim \Gamma(.01,.01)$. We write the normal distribution as $N(\mu,\tau)$ where $\mu$ is the mean and $\tau=1/\sigma^2$ is the precision. $\Gamma$ denotes the Gamma distribution.

    To obtain the necessary precision, please use run.jags defaults. Please don’t change options or modules.

    The last two questions belong to chapter 12 of the lecture! Remember that if $p$ is the probability of an event, then the odds are $o=\frac{p}{1-p}$.

    
    
    1. What is the lower boundary of the 95%-credible-interval for $\mu$?
    2. What is the upper boundary of the 95%-credible-interval for $\mu$?
    3. What is the lower boundary of the 95%-credible-interval for $\sigma$?
    4. What is the upper boundary of the 95%-credible-interval for $\sigma$?
    5. What are the posterior odds of $\mu>-9.48$?
    6. What are the posterior odds of $-10.4<\mu<-10.2$?
    
    

23. Question

    Use the data from the file DSQ.csv. You are interested in the interquartile range $\xi$ of the variable X2. The interquartile range is the distance between the 25% and 75% quantiles. In R you can use the function IQR(x) to determine the interquartile range of x.

    
    
    1. What is the plug-in estimate of the interquartile range of X2?
    2. Use a bootstrap (with 10000 replications) to determine the standard deviation of this estimate.
    
    

24. Question

    The file DGP.csv contains four variables, B, Q, X and p.

    The variable p denotes the group to which an observation belongs.

    You compare the following two models: A standard OLS model and a model with a random effect.

    • Here is the OLS model:

    $$X_{pt} = \beta_0 + \beta_B B_{pt} + \beta_Q Q_{pt} + \epsilon_{pt}$$

    • Now you extend this model with a random effect $\nu_{p}$:

    $$X_{pt} = \beta_0 + \beta_B B_{pt} + \beta_Q Q_{pt} + \nu_{p} + \epsilon_{pt}$$

    You use lmer from the lme4 library to estimate the model with random effects.

    
    
    1. What is your estimate for $\beta_{B}$ in the OLS model?
    2. What is your estimate for $\beta_{Q}$ in the OLS model?
    3. What is your estimate for $\beta_{B}$ in the model with a random effect?
    4. What is your estimate for $\beta_{Q}$ in the model with a random effect?
    5. What is your estimate for the standard deviation of the random effect $\nu_{p}$?