Lab 3

Simulating Data

\[ y_i = 0 + 5x_{i1} - 3x_{i2} + u_i \\ \quad u_i \sim N(0,1) \\ \boldsymbol{x}_i \sim N \left ([0,0], \begin{bmatrix} 1 & 0.5 \\ 0.5 & 1 \end{bmatrix} \right) \]

Write a function that takes as input a value for the number of simulations to execute, the sample size for each simulation, values for the three coefficients of the regression model above, and the correlation between \(x_1\) and \(x_2\). The function should do the following:

  1. Take \(N\) independent samples each from the distributions for \(\boldsymbol{x}_i\) and \(u_i\)
  2. Generate the model-implied values of \(y_i\) for each set of sampled values (given the assumed values of the three coefficients)
  3. Calculate the OLS regression coefficients (by “hand”) for the model using the simulated data you just generated
  4. Store the three estimated coefficients in a row of a matrix
  5. Repeat this process for the number of simulations requested, appending the matrix with the new row of coefficient estimates after each sim
  6. Return the final matrix of coefficient estimates for all simulations
# 0. what are the inputs of the function? 

# 1. what (canned) function can we use to take independent samples from a normal distribution?

# 2. what is the equation for the "model-implied" values of y_i?  

# 3. what is the (matrix multiplication) equation for the OLS coefficients? 

# 4. how do we store values into a new row (subset) of a matrix? 

# 5. what code do we use to repeat/iterate a process? 

# 6. what is the structure of the (output) object we want to initialize?

Unbiasedness

Execute your function for 100 simulations with a sample size of 100 for each simulation, storing the output in a new object called s1. Assume the coefficient values given above. Calculate the mean coefficient estimate across the simulations for each coefficient in the model. What do you find?

Do the same thing again, but use 1,000 simulations instead. What do you find?

# before you do this, what do you expect to find? 

# how do you expect the mean coefficient estimate across 100 simulations to differ from the mean coefficient estimate across 1000 simulations?

Calculating SE

  1. Simulate 100 observations for the model above
  2. Estimate an OLS regression for these 100 observations
  3. Calculate the standard errors for the three coefficients by hand

Hint: The standard errors are square root of the diagonal of the variance-covariance matrix

# 1. simulate 100 observations (steps 1 and 2 from above)

# 2. estimate an OLS regression (step 3 from above)

# 3. what are the steps for calculating the standard errors?

\[ Covariance \space Matrix = \sigma^2 (X'X)^{-1} \\ \sigma^2 \approx \frac{\hat{u}'\hat{u}}{N-K-1} \\ \hat{u_i} = y_i - X \hat{\beta} \]