Further Exercises for MW24.1 - 11

  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$X5) to determine the mean of the variable X5 in your data.
    4. Use the command median(DYQ$X5) to determine the median of the variable X5 in your data.
    5. Use the command sd(DYQ$X5) to determine the standard deviation of the variable X5 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 XX contains n=10n=10 independent and normally distributed observations: X1,,X10X_1,\ldots,X_{10}. You are looking for an estimator for E[X]E[X]. Which of the following statements are correct:

    1. The estimator 17X5+37X6+17X9+27X10\frac{1}{7}X_{5}+\frac{3}{7}X_{6}+\frac{1}{7}X_{9}+\frac{2}{7}X_{10} is an unbiased estimator for E[X]E[X].

      Yes / No

    2. The estimator 13X3+16X6+13X7112X8-\frac{1}{3}X_{3}+\frac{1}{6}X_{6}+\frac{1}{3}X_{7}-\frac{1}{12}X_{8} is an unbiased estimator for E[X]E[X].

      Yes / No

    3. The estimator X4X10X_{4}-X_{10} is an unbiased estimator for E[X]E[X].

      Yes / No

    4. The estimator 47X1+17X3+37X517X6\frac{4}{7}X_{1}+\frac{1}{7}X_{3}+\frac{3}{7}X_{5}-\frac{1}{7}X_{6} is an unbiased estimator for E[X]E[X].

      Yes / No

    5. The estimator X4+32X532X7X_{4}+\frac{3}{2}X_{5}-\frac{3}{2}X_{7} dominates 13X3+13X4+X8-\frac{1}{3}X_{3}+\frac{1}{3}X_{4}+X_{8}.

      Yes / No

    6. The estimator X4-X_{4} dominates 3X5+3X9+X10-3X_{5}+3X_{9}+X_{10}.

      Yes / No



  3. Question

    You use a level of significance of 0.001.


    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 XX follows a normal distribution with unknown mean and standard deviation 3. Your sample contains 24 observations. The sample mean is -9. Your Null-hypothesis is that XX has a mean of 20. How large is the absolute value of your test statistic?
    3. You still assume that the random variable XX follows a normal distribution with unknown mean and standard deviation 3. Now you consider a sample with 24 observations and sample mean 3. Your Null-hypothesis is still that XX has a mean of 20. How large is the pp-value (for a two-sided test, rounded to 4 decimal places)?

  4. Question

    Your data in the file DHC.csv contains two variables: T and p. The variable p tells you which group (B or M) the observation T belongs to.

    Compare the mean of T for two groups B and M with the help of a (two-sided) tt-test.

    1. Your Null-hypothesis is that the mean of the normally distributed T is the same in both groups. How large is the pp-value for this tt-test?
    2. You use a level of significance of 10%. Do you reject your Null-hypothesis? Yes / No


  5. Question

    The random variable XN(μ,σ2)X \sim N(\mu,\sigma^2) follows a normal distribution with unknown variance σ2\sigma^2. You draw a sample with 17 observations. You find a sample mean of 8 and a sample standard deviation of 2.

    • Determine a 95%-confindence interval for your estimation of the expected value of XX: μ̂\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 XX is distributed as follows:

    • P(X=A)=θP(X=A)=\theta
    • P(X=B)=θP(X=B)=\theta
    • P(X=C)=12θP(X=C)=1-2\theta

    We have θ[0,1/2]\theta\in[0,1/2].

    In your sample you have the following observations:

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

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


  7. Question

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

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

    Y=β0+β1X+uY = \beta_0 + \beta_1 X + u.

    1. Which value do you estimate for β1\beta_1?
    2. Your (two sided) Null-hypothesis is H0:β1=0H_0: \beta_1=0. Determine the pp-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 β1\beta_1?
    5. What is the upper boundary of the 95% confidence interval for β1\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 DSK.csv. Based on this data you estimate the following relationship:

    Y=β0+β1X1+β2X2+uY = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + u.

    1. Which value do you estimate for β2\beta_2?
    2. Your (two sided) Null-hypothesis is H0:β2=0H_0: \beta_2 = 0. Determine the pp-value for this test (report at least 4 decimal places).
    3. You use a level of significance of 5%. Can you reject your Null-hypothesis? Yes / No
    4. What is the lower boundary of the 95%-confidence-interval for β2\beta_2?
    5. What is the upper boundary of the 95%-confidence-interval for β2\beta_2?


  10. Question

    Consider the following model:

    pw=β0+β1ai+β2jb+β3aijb+upw = \beta_0 + \beta_1 \cdot ai + \beta_2 \cdot jb + \beta_3 \cdot ai \cdot jb + u

    The variable aiai indicates whether you are in situation BNM or DEQ: In the case of BNM you have ai=0ai=0. In the case of DEQ you have ai=1ai=1.

    The variable jbjb indicates whether you are in situation JXP or KFT: In the case of JXP you have jb=0jb=0. In the case of KFT you have jb=1jb=1.

    The mean values of pwpw for the four different combinations of BNM and DEQ and JXP and KFT are shown in the following table:


    1. What is β0\beta_0?
    2. What is β1\beta_1?
    3. What is β2\beta_2?
    4. What is β3\beta_3?

  11. Question

    Save the file DLE.csv in your working directory. You want to estimate the mean of the absolute value of X3.


    1. What is the plug-in estimate of the mean of the absolute value of X3?
    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 X3?

  12. Question

    The random variable LL follows a normal distribution with unknown mean μL\mu_{L} and known standard deviation σL=15\sigma_{L}=15.

    According to your prior the following holds:

    • μL=6\mu_{L}=6 with probability 1/3,
    • μL=16\mu_{L}=16 with probability 2/3.

    The probability for all other values of μL\mu_{L} is zero.

    You have one observation, L=2L=2.

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

    E.g. dnorm(2,6,15) yields the density of the normal distribution for L=2L=2 when μL=6\mu_{L}=6 and σL=15\sigma_{L}=15.


    1. What is the posterior probability P(μL=6|L=2)P(\mu_{L}=6| L=2 )?
    2. What is the posterior probability P(μL=11|L=2)P(\mu_{L}=11| L=2 )?
    3. Now you have two observations: L={2,15}L=\{2,15\}. What is the posterior probability P(μL=6|L={2,15})P(\mu_{L}=6| L=\{ 2,15 \} )?

  13. Question

    The file DMM.csv contains a variable X. This X is a sample of the random variable XX. Here we write the normal distribution as N(μ,τ)N(\mu,\tau) where μ\mu is the mean and τ=1/σ2\tau=1/\sigma^2 is the precision. You assume that XX follows a normal distribution: XN(μ,1/σ2)X \sim N(\mu,1/\sigma^2) where σ2\sigma^2 is the variance of XX. You have the following priors: μN(0,.0001)\mu \sim N(0,.0001), τ=1/σ2Γ(.01,.01)\tau=1/\sigma^2\sim \Gamma(.01,.01) (Γ\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 μ\mu?
    2. What is the upper boundary of the 95%-credible-interval for μ\mu?
    3. What is the lower boundary of the 95%-credible-interval for σ\sigma?
    4. What is the upper boundary of the 95%-credible-interval for σ\sigma?
    5. What is the posterior probability (a number between 0 and 1) of μ>1.26\mu>1.26?
    6. What is the posterior probability (a number between 0 and 1) of 0.629<μ<3.21-0.629<\mu<3.21?

  14. Question

    The file DGA.csv contains an independent variable B and a dependent binary variable H.

    You estimate the following model:

    P(H=1|B=b)=F(β0+βBb)P(H=1|B=b) = F(\beta_0 + \beta_B b)

    where FF is the logistic distribution.


    1. What is your estimate for βB\beta_B?
    2. What is the marginal effect of bb for the average value of BB in your data?
    3. What is the average marginal effect of bb?
    4. What is the marginal effect of bb if b=9b=9?
    5. What are the odds for H=1H=1 if b=9b=9?

  15. Question

    The file DST.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 θ\theta?
    4. Your Null-hypothesis is that θ=\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 pp-value do you get (rounded to 4 decimal places)?

  16. Question

    The file DBJ.csv contains three variables, j, W and Y. The variable j denotes to which group observations belong. Y is our dependent variable.

    We write the normal distribution as N(μ,τ)N(\mu,\tau) where μ\mu is the mean and τ=1/σ2\tau=1/\sigma^2 is the precision. Γ\Gamma denotes the Gamma distribution. You use JAGS to estimate the following model with random effects:

    Yjt=β0+νj+ϵjtY_{jt} = \beta_0 + \nu_{j} + \epsilon_{jt}

    where the group specific random effect νjN(0,τν)\nu_j \sim N(0,\tau_\nu) and the residual ϵjtN(0,τϵ)\epsilon_{jt} \sim N(0,\tau_\epsilon). Here τν=1/σν2\tau_\nu=1/\sigma^2_\nu and τϵ=1/σϵ2\tau_\epsilon=1/\sigma^2_\epsilon are the precision of νj\nu_j and ϵjt\epsilon_{jt}, respectively.

    Your priors are: β0N(0,0.0001)\beta_0 \sim N(0,0.0001), τνΓ(.01,.01)\tau_\nu \sim \Gamma(.01,.01), τϵΓ(.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 σϵ\sigma_\epsilon?
    2. What is the 50%-quantile of your posterior for σν\sigma_\nu?
    3. What is the posterior probability (a number between 0 and 1) of σν>2.6\sigma_\nu>2.6?

  17. Question

    The file DPV.csv contains the variables X, Y and Z. You estimate the model Y=β0+β1X+uY = \beta_0 + \beta_1 X + u.


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

  18. Question

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

    Y=β0+β1X1+β2X2+β3X3+β4X4+β5X5+β6X6+β7X7+uY = \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} + u


    1. What is the coefficient of X4 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 X4. What is now the coefficient of X4?
    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 X4 from the model, add X4 back to your model. What is the coefficient of X4 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 J and a dependent binary variable T.

    You estimate the following model:

    P(T=1|[J=j)=Φ(β0+βjj) P(T=1|[J=j) = \Phi(\beta_0 + \beta_{j} j)

    where Φ\Phi is the standard normal distribution.


    1. What is your estimate for βj\beta_{j}?
    2. What is the marginal effect of jj for the average value of jj in your data?
    3. What is the average marginal effect of jj?
    4. What is the marginal effect of jj if j=1.504j=-1.504?

  20. Question

    A random variable XX follows a distribution with density function f(x|θ)=(x24)θθxf(x|\theta)=\left(\frac {x}{24}\right)^\theta \cdot \frac{\theta}{x} if x[0,24]x\in[0,24] and f(x)=0f(x)=0 otherwise.

    Your sample contains the observations {8,8,9,11,13}\{ 8, 8, 9, 11, 13 \}.

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


  21. Question

    The data in the file DGX.csv contains 8 variables, Ynd, Ytv, Yue, Ywc, Xnd, Xtv, Xue, Xwc. You investigate the effect Xnd has on Ynd, the effect Xtv has on Ytv, the effect Xue has on Yue, the effect Xwc has on Ywc. For each case below, select the most suitable specification and provide the point estimate of the effect.

    1. Use a specification where Ynd changes by a fixed number of percentage points when Xnd changes by one unit. By how many percentage points does Ynd change approximately when Xnd changes by one unit?
    2. Use a specification where Ywc changes by a fixed amount when Xwc changes by a given percentage. By which amount does Ywc change when Xwc changes by 1 percentage point?
    3. Use a specification where the elasticity of Ytv with respect to Xtv is constant. What is the elasticity of Ytv with respect to Xtv?
    4. Use a specification where the marginal effect of Xue on Yue is constant. What is the marginal effect of Xue on Yue?


  22. Question

    The file DTK.csv contains a variable X. This X is a sample of the random variable XX. You assume that XX follows a normal distribution: XN(μ,1/σ2)X \sim N(\mu,1/\sigma^2) where σ2\sigma^2 is the variance of XX. Your priors are μN(14.9,0.7)\mu \sim N(-14.9,0.7), τ=1/σ2Γ(.01,.01)\tau=1/\sigma^2\sim \Gamma(.01,.01). We write the normal distribution as N(μ,τ)N(\mu,\tau) where μ\mu is the mean and τ=1/σ2\tau=1/\sigma^2 is the precision. Γ\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 pp is the probability of an event, then the odds are o=p1po=\frac{p}{1-p}.


    1. What is the lower boundary of the 95%-credible-interval for μ\mu?
    2. What is the upper boundary of the 95%-credible-interval for μ\mu?
    3. What is the lower boundary of the 95%-credible-interval for σ\sigma?
    4. What is the upper boundary of the 95%-credible-interval for σ\sigma?
    5. What are the posterior odds of μ>13.7\mu>-13.7?
    6. What are the posterior odds of 13.7<μ<13-13.7<\mu<-13?

  23. Question

    Use the data from the file DBK.csv. You are interested in the interquartile range ξ\xi of the variable X1. 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 X1?
    2. Use a bootstrap (with 10000 replications) to determine the standard deviation of this estimate.

  24. Question

    The file DGP.csv contains four variables, E, P, S and b.

    The variable b 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:

    Sbt=β0+βEEbt+βPPbt+ϵbtS_{bt} = \beta_0 + \beta_E E_{bt} + \beta_P P_{bt} + \epsilon_{bt}

    • Now you extend this model with a random effect νb\nu_{b}:

    Sbt=β0+βEEbt+βPPbt+νb+ϵbtS_{bt} = \beta_0 + \beta_E E_{bt} + \beta_P P_{bt} + \nu_{b} + \epsilon_{bt}

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


    1. What is your estimate for βE\beta_{E} in the OLS model?
    2. What is your estimate for βP\beta_{P} in the OLS model?
    3. What is your estimate for βE\beta_{E} in the model with a random effect?
    4. What is your estimate for βP\beta_{P} in the model with a random effect?
    5. What is your estimate for the standard deviation of the random effect νb\nu_{b}?