A homework on mulitway fixed effect approach

In this homework we are going to consider a fictuous grading at a given universities and try to see what we can learn from grade variability and it affects earnings.

In the first data set we have the grades collected by a bunch of student within a given summester. Each student attended 2 courses. From this data we are going to try to extrac the ability of each student while allowing for course specific intercept. We can then use this to evaluate how much of the grade variation is due to student differences versus course differences and other factors (residuals).

Given this ability measures, we then merge a second file which has the earnings of the student at age 35. We then evaluate the effect of academic ability on individual earnings. Here again we will worry about the effect of over-fitting.

Of course this requires, like we saw in class, estimating many parameters, hence we will look into overfitting and how to address it! We wil lmake use of sparse matrices, degree of freedom correction and bias correction.

The two data files you will need are:

• grades: hw4-grades.json
• earnings: hw4-earnings.json

Useful links: - Sparse linear solver

import os
import pandas as pd
import tqdm
import seaborn as sns
import matplotlib.pyplot as plt
import numpy as np
%cd ..
%load_ext autoreload
%autoreload 2

/home/tlamadon/git/econ-21340

import solutions.sol_pset4 as solution # you need to command this, you don't have the solution file!
#solution.simulate_data()

Explaining the dispersion in grades

Load the grade data from hw4-grades.json. Then compute:

1. total variance in grades
2. the between class variance
3. plot the histogram of class size

df_all = solution.question1()

ns = len(np.unique(df_all['student_id']))
nc = len(np.unique(df_all['class_id']))
nn = len(df_all)

print("{} students and {} classes".format(ns,nc))

df_all[["grade","class_id","student_id","major","firstname"]]

 total variance is 2.1867408049173007
 between class variance is 0.7541357576225806

6732 students and 673 classes

	grade	class_id	student_id	major	firstname
0	-3.099938	MQ2281	5	SOCIAL SCIENCE OR HISTORY TEACHER EDUCATION	Francis
1	-3.021075	JP1499	5	SOCIAL SCIENCE OR HISTORY TEACHER EDUCATION	Francis
2	-0.042223	HO0525	20	ELEMENTARY EDUCATION	Cruz
3	-1.842850	BZ6573	20	ELEMENTARY EDUCATION	Cruz
4	0.038426	FM1995	21	MISCELLANEOUS PSYCHOLOGY	Denise
...	...	...	...	...	...
13459	-2.082608	JH2331	50684	COMMUNICATIONS	Lavera
13460	-1.478929	IE5546	50687	COGNITIVE SCIENCE AND BIOPSYCHOLOGY	Garold
13461	-0.179943	MG0664	50687	COGNITIVE SCIENCE AND BIOPSYCHOLOGY	Garold
13462	5.874171	CA0214	50693	GENERAL BUSINESS	Gus
13463	4.786490	JA7623	50693	GENERAL BUSINESS	Gus

13464 rows × 5 columns

Constructing the sparse regressor matrices

In a similar fashion to what we covered in the class we want to estimate a two-way fixed model of grates. Specifically, we are want to fit:

$$y_{ic} = \alpha_i + \psi_c + \epsilon_{ic}$$

where $i$ denotes each individual, $c$ denote each courses and $\epsilon_{ic}$ is an error term that will assume conditional mean independent of the assignment of students to courses.

We are going to estimate this using least-square. This requires that we construct the matrices that correspond to the equation for $y_{ic}$. We then want to consruct the $A$ and $J$ such that

$$Y = A \alpha + J \psi + E$$

where for $n_s$ students each with $n_g$ grades in difference courses and a total of $n_c$ courses we have that $Y$ is $n_s \cdot n_g \times 1$ vector, $A$ is a $n_s \cdot n_g \times n_s$ matrix and $J$ is $n_s \cdot n_g \times n_c$. $\alpha$ is the vector of size $n_s$ and $\psi$ is a vector of size $n_c$.

Each fo the $n_s \cdot n_g$ correspond to a grade, in each row $A$ has a $1$ in the column corresponding to the individual of this row. Similary, $J$ has a $1$ for for the column corresponding to the class of that row.

So, I ask you to:

1. construct these matrices using python sparse matrices scipy.sparse.csc.csc_matrix

from scipy.sparse.linalg import spsolve
Y,A,J = solution.question2(df_all)

print(type(A))
print(A.shape)
print(J.shape)
print(Y.shape)
print(A.sum())
print(J.sum())

# getting a nice diagonal here requires sorting by the studend_id
plt.spy(A,aspect=0.1,markersize=0.2)
plt.show()
plt.spy(J,aspect=0.1,markersize=0.2)
plt.show()

<class 'scipy.sparse.csc.csc_matrix'>
(13464, 6732)
(13464, 673)
(13464,)
13464.0
13464.0

Estimating the model

Next we estimate our model using the OLS estiamtor formula. We first remove the last column of $J$ (since the model we wrote does not pin down a constant we force the last course to have $\psi=0$). Solve the linear system using the formula

$$\hat{\gamma} = (M'M)^{-1} M' Y$$

where $M = [A,J]$ and $\gamma = (\alpha,\psi)$.

So do the following:

1. select the last column simply by doing J = J[:,1:(nc-1]]
2. use scipy.sparse.hstack to concatenate the matrices to create M
3. use scipy.sparse.linalg.spsolve to solve a sparse linear system
4. extract $\hat{\alpha}$ from $\hat{\gamma}$ by selecting the first $n_s$ terms
5. merge $\hat{\alpha}$ into df_all
6. compute the variance of $\hat{\alpha}$ in df_all
7. compute the variance of the residuals
8. What share of the total variation in grades can be attributed to difference in students?

df_all, M, gamma_hat = solution.question3(df_all,A,J,Y)
print(df_all['alpha_hat'].var(),df_all['resid'].var(),df_all['grade'].var())
df_all[["grade","class_id","student_id","major","firstname","alpha_hat"]]

1.347920639414136 0.2528814610764115 2.1867408049173007

	grade	class_id	student_id	major	firstname	alpha_hat
0	-3.099938	MQ2281	5	SOCIAL SCIENCE OR HISTORY TEACHER EDUCATION	Francis	-3.626313
1	-3.021075	JP1499	5	SOCIAL SCIENCE OR HISTORY TEACHER EDUCATION	Francis	-3.626313
2	-0.042223	HO0525	20	ELEMENTARY EDUCATION	Cruz	-2.322520
3	-1.842850	BZ6573	20	ELEMENTARY EDUCATION	Cruz	-2.322520
4	0.038426	FM1995	21	MISCELLANEOUS PSYCHOLOGY	Denise	-1.108136
...	...	...	...	...	...	...
13459	-2.082608	JH2331	50684	COMMUNICATIONS	Lavera	-2.349589
13460	-1.478929	IE5546	50687	COGNITIVE SCIENCE AND BIOPSYCHOLOGY	Garold	-1.186850
13461	-0.179943	MG0664	50687	COGNITIVE SCIENCE AND BIOPSYCHOLOGY	Garold	-1.186850
13462	5.874171	CA0214	50693	GENERAL BUSINESS	Gus	3.840780
13463	4.786490	JA7623	50693	GENERAL BUSINESS	Gus	3.840780

13464 rows × 6 columns

A simple evaluation of our estimator

To see what we are dealing with, we are simly going to re-simulate using our estimated parameters, then re-run our estimation and compare the new results to the previous one. This is in the spirit of a bootstrap exercise, onyl we will just do it once this time.

Please do:

1. create $Y_2 = M \hat{\gamma} + \hat{\sigma}_r E$ where $E$ is a vector of draw from a standard normal.
2. estimate $\hat{\gamma}_2$ from $Y_2$
3. report the new variance term and compare them to the previously estimated
4. comment on the results (not that because of the seed and ordering, your number doesn't have to match mine exactly)

df_all = solution.question4(df_all,M,gamma_hat)
df_all['alpha_hat2'].var(),df_all['resid2'].var()

(1.4806557269553007, 0.11529175149045187)

Noticed how even the variance of the residual has shrunk? Now is the time to remember STATS 101. We have all heard this thing about degree of freedom correction! Indeed we should correct our raw variance estimates to control for the fact that we have estimated a bunhc of dummies. Usually we use $n/n-1$ because we only estimate one mean. Here however we have estimated $n_s +n_c - 1$ means! Hence we should use

$$\frac{N}{N-n_s -n_c +1} \hat{Var}(\hat{\epsilon})$$

please do:

1. compute this variance corrected for degree of freedom using your recomputed residuals
2. compare this variance to the variance you estimated in quetion 3
3. what does this suggest about your estimates in Q3?

solution.question4b(df_all)

Var2 with degree of freedom correction:0.256153158757004

Evaluate impact of academic measure on earnings

In this section we load a separate data set that contains for each student their earnings at age 35. We are intereted in the effect of $\alpha$ on earnings.

Do the following:

1. load the data the earnings data listed in the intro
2. merge $\alpha$ into the data
3. regress earnings on $\alpha$.

df_earnings = solution.question5(df_all)
df_earnings

                            OLS Regression Results                            
==============================================================================
Dep. Variable:               earnings   R-squared:                       0.035
Model:                            OLS   Adj. R-squared:                  0.035
Method:                 Least Squares   F-statistic:                     245.3
Date:                Mon, 20 Sep 2021   Prob (F-statistic):           2.52e-54
Time:                        11:49:03   Log-Likelihood:                -9663.9
No. Observations:                6732   AIC:                         1.933e+04
Df Residuals:                    6730   BIC:                         1.935e+04
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept      0.1267      0.014      8.895      0.000       0.099       0.155
alpha_hat      0.1672      0.011     15.661      0.000       0.146       0.188
==============================================================================
Omnibus:                        0.805   Durbin-Watson:                   2.014
Prob(Omnibus):                  0.669   Jarque-Bera (JB):                0.843
Skew:                          -0.016   Prob(JB):                        0.656
Kurtosis:                       2.956   Cond. No.                         1.86
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

	student_id	major	firstname	earnings	alpha_hat
0	5	SOCIAL SCIENCE OR HISTORY TEACHER EDUCATION	Francis	-1.271396	-3.626313
1	20	ELEMENTARY EDUCATION	Cruz	0.227663	-2.322520
2	21	MISCELLANEOUS PSYCHOLOGY	Denise	0.011602	-1.108136
3	28	MATHEMATICS TEACHER EDUCATION	Kurtis	0.998318	1.906966
4	30	N/A (less than bachelor's degree)	Flor	0.645380	0.850472
...	...	...	...	...	...
6727	50676	ENGINEERING MECHANICS PHYSICS AND SCIENCE	Angelia	1.501068	0.433212
6728	50681	ELECTRICAL ENGINEERING TECHNOLOGY	Stacy	0.954807	-0.551522
6729	50684	COMMUNICATIONS	Lavera	-1.701903	-2.349589
6730	50687	COGNITIVE SCIENCE AND BIOPSYCHOLOGY	Garold	0.114382	-1.186850
6731	50693	GENERAL BUSINESS	Gus	1.682317	3.840780

6732 rows × 5 columns

Bias correction - construct the Q matrix

We want to apply bias correction to refine our results. As we have seen in class thaqt we can directly compute the bias of the expression of interest.

$$E[ \hat{\gamma} Q \hat{\gamma}' ] = \gamma Q \gamma + \frac{\sigma^2}{n} \text{Tr}[ ( M'M )^{-1} Q]$$

under homoskedatic assumption of the error and hence we get the following expresison for the bias for any $Q$ matrix:

$$B = \frac{\sigma^2}{N} \text{Tr}[ ( M'M )^{-1} Q]$$

When computing the variance of the measured ability of the student, we simply use a diagonal matrix on $\gamma$ which selects only the ability part and removes the average.

do: 1. Construct such Q matrix. 2. check that $\gamma Q \gamma' = \hat{Var}(\hat{a})$.

# a small example if we had ns=5,nc=4
Qbis = solution.question6(ns=5,nc=4)
print(Qbis)

# the full Q
Q = solution.question6(ns,nc)

# comparing Q expression to df_all expression
1/(ns)*np.matmul( gamma_hat, np.matmul(Q,gamma_hat )), gamma_hat[range(ns)].var()

[[ 0.8 -0.2 -0.2 -0.2 -0.2  0.   0.   0. ]
 [-0.2  0.8 -0.2 -0.2 -0.2  0.   0.   0. ]
 [-0.2 -0.2  0.8 -0.2 -0.2  0.   0.   0. ]
 [-0.2 -0.2 -0.2  0.8 -0.2  0.   0.   0. ]
 [-0.2 -0.2 -0.2 -0.2  0.8  0.   0.   0. ]
 [ 0.   0.   0.   0.   0.   0.   0.   0. ]
 [ 0.   0.   0.   0.   0.   0.   0.   0. ]
 [ 0.   0.   0.   0.   0.   0.   0.   0. ]]





(1.34782052647301, 1.3478205264730092)

Bias correction - Variance share

We are now finally in the position to compute our bias. We have are matrix $Q$. Now we also need the variance of the residual! Given what we have learn in Question 4, we definitely want to use the formula with degree of freedom correction.

1. Compute $\sigma^2_r$ with the degree of freedom correction
2. Invert $M'M$ using scipy.sparse.linalg
3. Compute $B = \frac{\sigma^2}{N} \text{Tr}[ ( M'M )^{-1} Q]$ using np.trace
4. Remove this from original estimate to get the share of variance explained by student differences!

Note that inversing a matrix is far longer than solving a linear system. You might need to be patient here!

B = solution.question7(M,Q,df_all)
B

/home/tlamadon/git/econ-21340/envs/lib/python3.8/site-packages/scipy/sparse/linalg/dsolve/linsolve.py:318: SparseEfficiencyWarning: splu requires CSC matrix format
  warn('splu requires CSC matrix format', SparseEfficiencyWarning)
/home/tlamadon/git/econ-21340/envs/lib/python3.8/site-packages/scipy/sparse/linalg/dsolve/linsolve.py:215: SparseEfficiencyWarning: spsolve is more efficient when sparse b is in the CSC matrix format
  warn('spsolve is more efficient when sparse b '





0.3119489890374221

gamma_hat[range(ns)].var() - B

1.0358715374355871

Bias correction - Regression coefficient

Finally, we look back at our regression of earnings on estimated academic ability. We have seen in class that when the regressor has measurment error this will suffer from attenuation bias. Here we now know exactly how much of the variance is noise thanks to our bias correction.

The attenuation bias is given by :

$$\beta_2 = \frac{Var(x)}{Var(x) + B} \beta$$

We then decide to compute a correction for our regression using our estimated $B$. This means computing a corrected parameters as follows:

$$\hat{\beta}^{BC} = \frac{Var(x) + B}{Var(x)} \hat{\beta}^{Q5} = \frac{Var(\hat{\alpha})}{Var(\hat{\alpha})-B} \hat{\beta}^{Q5}$$

Do:

1. compute the corrected $\hat{\beta}^{BC}$
2. FIY, the true $\beta$ I used to simulate the data was 0.2, is your final parameter far? Is is economically different from the $\hat{\beta}^{Q5}$ we got in Question 5?

Conclusion

I hope you have learned about the pitfalls of over-fitting in this assignment! There are many and they can drastically affect the results and the conclusion of an empirical analysis.

This is the end of the class, I hope you enjoyed it and that you learned a thing or twom, have a nice summer!