## Preamble: Package Loading
import numpy as np
import ipywidgets as ipw
from IPython.display import display,display_html
import matplotlib.pyplot as plt
from matplotlib import gridspec
import pandas as pd
import json
import kernel as kr
import psc_dbl_sumdisp as psd
%%html
<style>
#.cell.selected~.unselected { display: none; }
#.cell.code_cell.unselected .input { display: none; }
</style>
Selection of Heterogenous Instruments in Partially Linear Fixed Effect Panel Regression
Monte Carlo Study Results
By: Eric Penner
Summary
The following notebook contains results of a Monte Carlo Exercise conducted on the estimator detailed in 'shi.ipynb' and 'shi_wp.pdf' with a data sets generated by 'shi_dgp.ipnyb' (see this notebook for details of the DGP).
Important features of each of the following trials are presented here
- All estimates have been generated with the knowledge that the secondary equations are panel type (i.e. the estimation of the secondary equations is properly specified).
- The number of datasets used from each component of each trial is 'nds = 500'
# Trial 1.1: 0
inpt_filenames=[['pscout_9_4_1347.json' ,'pscout_9_4_1505.json' ,'pscout_9_4_1913.json' ,'pscout_9_4_1816.json']]
line_nms=[['Oracle','Known','Unknown','Lasso']]
#Trial 1.2: 1
inpt_filenames.append(['pscout_9_4_1232.json' ,'pscout_9_4_1837.json','pscout_9_4_1969.json' ,'pscout_9_4_1655.json' ])
line_nms.append(['Oracle','Known','Unknown','Lasso'])
#Trial 1.3: 2
inpt_filenames.append(['pscout_9_4_1636.json','pscout_9_4_1191.json','pscout_9_4_1510.json'])
line_nms.append(['Oracle','Known','Lasso'])
#Trial 1.4: 3
inpt_filenames.append(['pscout_9_4_1816.json','pscout_9_4_1655.json','pscout_9_4_1510.json','pscout_9_5_1624.json'])
line_nms.append(['t_inst=15','t_inst=30','t_inst=45','t_inst=100'])
#Trial 2.0: 4
inpt_filenames.append(['pscout_9_5_1624.json','pscout_9_6_1331.json','pscout_9_6_1606.json'])
line_nms.append(['ncs = 10','ncs = 25','ncs = 40'])
#Trial 3.0: 5
inpt_filenames.append(['pscout_9_5_1624.json','pscout_9_6_1537.json','pscout_9_6_1725.json'])
line_nms.append(['ntp = 50','ntp = 100','ntp = 150'])
res_out = [[psd.psc_load(inpt_filenames[k][i]) for i in range(len(inpt_filenames[k]))]
for k in range(len(line_nms))]
estin_dcts = [[res_out[k][i][0] for i in range(len(inpt_filenames[k]))]
for k in range(len(line_nms))]
dgp_sum_filenames = [[estin_dcts[k][i]['input_filename'].replace('pscdata','pscsum')
for i in range(len(inpt_filenames[k]))]
for k in range(len(line_nms))]
dgp_dicts = [[psd.pscsum_load(dgp_sum_filenames[k][i])
for i in range(len(dgp_sum_filenames[k]))]
for k in range(len(line_nms))]
dgpin_dcts = [[dgp_dicts[k][i][0] for i in range(len(inpt_filenames[k]))]
for k in range(len(line_nms))]
merged_dcts = [[{**estin_dcts[k][i],**dgpin_dcts[k][i]}
for i in range(len(inpt_filenames[k]))]
for k in range(len(line_nms))]
true_bcoeffs = [[dgp_dicts[k][i][1] for i in range(len(inpt_filenames[k]))]
for k in range(len(line_nms))]
true_acoeffs = [[dgp_dicts[k][i][2] for i in range(len(inpt_filenames[k]))]
for k in range(len(line_nms))]
bcoeff = [[res_out[k][i][1] for i in range(len(inpt_filenames[k]))]
for k in range(len(line_nms))]
acoeff = [[res_out[k][i][3] for i in range(len(inpt_filenames[k]))]
for k in range(len(line_nms))]
btables = [[res_out[k][i][2] for i in range(len(inpt_filenames[k]))]
for k in range(len(line_nms))]
atables = [[res_out[k][i][4] for i in range(len(inpt_filenames[k]))]
for k in range(len(line_nms))]
DGP: Base Model
For each time period $t \in \{1,2, \ldots, T\}$, component $ d\in \{1,2, \ldots , p_1\}$ of $Z_{1jt}$, and cross-section $j \in \{1,2,\ldots, q\}$ where $\{q,T\} \in \mathbb{N}$ consider the following,
\begin{align} Y_{jt} &= \beta_0 + [\; Z_{1jt}' \;\; Z_{2jt}' \;] \beta_1 + e_j + \varepsilon_{jt} \\[1em] % Z_{1jt,d} &= \alpha_{0jd} + Z_{2jt}' \alpha_{1jd} + W_{jt}' \alpha_{2jd} + V_{jt,d} \tag{2d} \\[1em] % E(&\varepsilon_{jt} | Z_{1jt} , Z_{2jt}) = E(\varepsilon_{jt} | Z_{1jt}) \neq 0 \\[1em] % E(& V_{jdt} | Z_{1j(t-1)},\ldots,Z_{1j(t-c)},Z_{2jt},W_{jt}) = 0 % \end{align}
DGP: Error, Instrument, and Exogenous Variable Generation
Let; $n_{tp} \equiv T$ be the total number of time periods, $n_{end} \equiv p_1$ be the number of endogneous regressors included in the primary regression, $n_{exo} \equiv p_2$ be the number of exogenous regressors included in the primary regression, and $n_{tinst} \equiv w$, $ n_{cinst} \equiv w_j$ be the total number of available instruments and the number of instruments relevant to each crossection respectively. Now let,
$$ \begin{align*} \rho_{er} &= \begin{bmatrix} \rho_{er,1} & \rho_{er,2} & \cdots & \rho_{er,n_{end}} \end{bmatrix} \\ \rho_{inst} &= \begin{bmatrix} \rho_{inst,1} & \rho_{inst,2} & \cdots & \rho_{inst,n_{inst}-1} \end{bmatrix}\\ \rho_{ex} &= \begin{bmatrix} \rho_{ex,1} & \rho_{ex,2} & \cdots & \rho_{ex,n_{ex}-1} \end{bmatrix} \end{align*} $$
So that I can define the covariance matrices for each cross section as follows
$$ \begin{align*} cv_{er} &= \begin{bmatrix} 1 & \rho_{er,1} & \rho_{er,2} & \cdots & \rho_{er,n_{end}} \\ \rho_{er,1} & 1 & \rho_{er,1} &\cdots & \rho_{er,n_{end}-1} \\ \rho_{er,2} & \rho_{er,1} & 1 & \cdots & \rho_{er,n_{end}-2} \\ \vdots & &&\ddots& \\ \rho_{er,n_{end}} & \rho_{er,n_{end}-1} & \rho_{er,n_{end}-2} & \cdots & 1 \end{bmatrix} % \hspace{1cm} % % cv_{ex} &= \begin{bmatrix} 1 & \rho_{ex,1} & \rho_{ex,2} & \cdots & \rho_{ex,n_{ex}-1} \\ \rho_{ex,1} & 1 & \rho_{ex,1} &\cdots & \rho_{ex,n_{ex}-2} \\ \rho_{ex,2} & \rho_{ex,1} & 1 & \cdots & \rho_{ex,n_{ex}-3} \\ \vdots & &&\ddots& \\ \rho_{ex,n_{ex}-1} & \rho_{ex,n_{ex}-2} & \rho_{ex,n_{ex}-2} & \cdots & 1 \end{bmatrix} \end{align*} $$
Since all crossections will use some subset of the vector of instruments the following is the full covariance matrix for all instruments.
$$ CV_{inst} = \begin{bmatrix} 1 & \rho_{inst,1} & \rho_{inst,2} & \cdots & \rho_{inst,n_{tinst}-1} \\ \rho_{inst,1} & 1 & \rho_{inst,1} &\cdots & \rho_{inst,n_{tinst}-2} \\ \rho_{inst,2} & \rho_{tinst,1} & 1 & \cdots & \rho_{inst,n_{tinst}-3} \\ \vdots & &&\ddots& \\ \rho_{inst,n_{tinst}-1} & \rho_{inst,n_{tinst}-2} & \rho_{inst,n_{tinst}-3} & \cdots & 1 \end{bmatrix} % $$
As a result we can construct the covariance matrices for all cross sections,
$$ \begin{align*} CV_{er} &= \begin{bmatrix} cv_{er} & \mathbf{0}_{(n_{end}+1 \times n_{end}+1)} & \cdots & \mathbf{0}_{(n_{end}+1 \times n_{end}+1)} \\ \mathbf{0}_{(n_{end}+1 \times n_{end}+1)} & cv_{er} & \cdots & \mathbf{0}_{(n_{end}+1 \times n_{end}+1)} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0}_{(n_{end}+1 \times n_{end}+1)} & \mathbf{0}_{(n_{end}+1 \times n_{end}+1)} & \cdots & cv_{er} \end{bmatrix} % \hspace{1cm} % CV_{ex} = \begin{bmatrix} cv_{ex} & \mathbf{0}_{(n_{ex} \times n_{ex})} & \cdots & \mathbf{0}_{(n_{ex} \times n_{ex})} \\ \mathbf{0}_{(n_{ex} \times n_{ex})} & cv_{ex} & \cdots & \mathbf{0}_{(n_{ex} \times n_{ex})} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0}_{(n_{ex} \times n_{ex})} & \mathbf{0}_{(n_{ex} \times n_{ex})} & \cdots & cv_{ex} \end{bmatrix} \end{align*} $$
Now I generate, error terms, instruments, and exogneous variables from mulitvariate normal distribution. First let
$$ \begin{align*} Z_{2jt} &= \begin{bmatrix} Z_{2jt,1} & Z_{2jt,2} & \cdots & Z_{2jt,n_{ex}} \end{bmatrix}' \\[10pt] W_t &= \begin{bmatrix} W_{t,1} & W_{t,2} & \cdots & W_{t,n_{inst}} \end{bmatrix}' \\[10pt] \tilde{V}_{jt} &= \begin{bmatrix} V_{jt,1} & V_{jt,2}& \cdots & V_{jt,n_{end}} & \varepsilon_{j} \end{bmatrix}' \end{align*} $$
Then consider, $ W_{t} \sim N(\mathbf{0}_{n_{inst} \times 1}, CV_{inst})$
$$ \begin{bmatrix} Z_{21t}' & Z_{22t}' & \cdots & Z_{2n_{cs}t}' \end{bmatrix}' \sim N(\mathbf{0}_{n_{cs} \cdot n_{exo} \times 1}, CV_{ex}) \hspace{1cm} \text{ and } \hspace{1cm} \begin{bmatrix} \tilde{V}_{1t}' & \tilde{V}_{2t}' & \cdots & \tilde{V}_{n_{cs},t}' \end{bmatrix}' \sim N(\mathbf{0}_{n_{cs} \cdot (n_{end} +1) \times 1}, CV_{er}) $$
DGP: Endogenous Variable Generation:
I will generate endogenous variables $ Z_{1jd} $ according to the following steps.
- For each $d \in \{1,2,\cdots,n_{end} \}$ I will draw the coefficienct vector $\alpha_{1d}$ from the cartesian product of $[1,1]$ with itself $n_{exo}$ times.
- For each $j\in \{1,2,\cdots , n_{cs}\}$ I will draw a sequence of integers $C_{j}$ from the $\mathcal{C}^{n_{tinst}}_{n_{cinst}}$ ways that that you can choose $n_{cinst}$ instruments from $n_{tinst}$ total instrument to be included in every regression of $Z_{1j}$ on $Z_{2j}$ and $W$ i.e. these numbers define which columns of $W$ are included in $W_j$.
- For each $d \in \{1,2,\cdots,n_{end} \}$ I will draw the coefficienct vector $\alpha_{2d}$ from the cartesian product of $[1,1]$ with itself $n_{tinst}$ times.
- Then for each $j\in \{1,2,\cdots , n_{cs}\}$ and $d \in \{1,2,\cdots,n_{end} \}$ I generate endogenous regressors $Z_{1jd}$ as follows
$$ Z_{1jd} = \alpha_{0jd} + Z_{2jt}' \alpha_{1d} + W_{jt}' \alpha_{2d} + V_{jt,d} \hspace{1cm} \text{ where } \hspace{1cm} \alpha_{0jd} = 1/2+j/2 $$
DGP: Primary Regressand Generation
Having generated all primary regressors I will generate the regresand for the primary equation to do this I will,
- I will draw the coefficienct vector $\beta_1$ from the cartesian product of $[1,1]$ with itself $n_{exo} +n_{end}$ times. Then set
$$ Y_{jt} = [\; Z_{1jt}' \;\; Z_{2jt}' \;] \beta_1 + e_j + \varepsilon_{jt} \;\;\;\; \text{ where } \;\;\;\; e_{j} = 1+j/2 $$
Variable Description Table
A number of variables are used below, here are their descriptions. Refer back to 'psc.ipynb' or 'psc_dgp.ipynb' for more details.
| Variable Name | Description |
|---|---|
| k_H | Kernel number used for H function Estimation |
| c_H | Plug in bandwidth constant for H function Estimation |
| k_mvd | Kernel number used for multivariate d>2 density estimation |
| c_mvd | Plug in bandwidth constant for multivariate d>2 density estimation |
| k_uvd | Kernel number used for bivariate density estimation |
| c_uvd | Plug in bandwidth used for bivariate density estimation |
| dep_nm | Variable name of the dependent variable |
| en_nm | Variable names of each endogenous variabble |
| ex_nm | Variable names of each exogenous variable |
| in_nm | Variable names of instruments relevant to each cross section |
| err_vpro | Vector of covariances used to construct the error cov matrix |
| ex_vpro | Vector of covariances used to construct the exog variable cov matrix |
| inst_vpro | Vector of covariances used to construct the instrument cov matrix |
| frc | Indicator for whether the functional form of control function is forced |
| input_filename | Filename of dataset used to generate the results. |
| kwnsub | Indicator for ifthe subset of instrument relevant to each crs is known |
| n_end | Number of endogenous variables |
| n_exo | Number of exogenous variables |
| ncs | Number cross sections |
| nds | Number of dgp data sets |
| ntp | Number of time periods |
| orcl | Indicator for whether residuals $V$ are observed (=1) or not |
| r_seed | Random number generator seed used to generate the data set |
| sec_pan | Indicator for whether the secondary eqn data is panel or not |
| c_inst | Number of instrument relevant to each cross section |
| t_inst | Total number of instruments |
| inc | List of instrument relevant to at least one cross section |
| tin | Variable name of the time period index |
| cin | Variable name of the cross section index |
| lasso | Indicator for lasso estimation |
| alph | lasso penalty value |
| epsil | Threshold for averaging "non zero" coefficients |
Trial Set 1: Estimator comparison by varying the total number of instruments
Trial Set 1.1: Estimator Comparison when $t_{inst} = 15$
Here we examine the sampling distribution of $\hat{\beta}_1$, and $ \hat{\alpha}_{1} $.
- Number of Cross Sections: 10
- Number of Endogenous Regressors: 1
- Number of Exogenous Regressors: 1
- Total Number of Instruments: 15
- Number of Instrument Relevant to Each Cross Section: 3
Trial Set 1.1: Merged DGP and Estimator Function Input Dictionary Comparison
Here I have merged together the dictionaries used to generate both the underlying dataset and the results (you will see the file name for this data set below) and the dictionary used to produce the estimates based on that data below. For a description of each variable name please see Variable Description
merged_dcts[0][0]
Trial Set 1.1: True Secondary Equation Coefficients Comparison
Here I display the coefficent vectors $\alpha_{1jd}$ used to generate the data set (by row indicating cross section and equation) corresponding to the position its file name appears in 'input_filenames0' above.
Note:
1.) A zero coefficient in the following matrix means that the instrument it multiplies is not relevant to that cross section.
2.) The density of the secondary regression coefficient matrix is 25%
display_html(true_acoeffs[0][0][0])
Trial Set 1.1: Secondary Function Coefficient Estimates
Here I show characteristics of the sampling distributions of the elements of $\hat{\alpha}_{dj}$.
for i in range(1,len(atables[0])):
display(atables[0][i][0])
display(line_nms[0][i])
Trial Set 1.1: Comments on Secondary Function Coefficient Estimates
- Here you can see that, as we would expect, the known subset estimator generates unbiased estimators, the unknown subset estimator generates badly biased estimates, and the lasso estimator generates estimates whose bias roughly splits that difference between the previous two.
- Note that the properties of the lasso estimator are what you would expect from a consistent estimator, where the variance of the sampling distribution of each coefficient in inversely proportional to the number of crossections that the instrument it multiplies is relevant to.
- This presentation is really only a frame of reference as the sampling distribution of the estimates isn't of first order importance in this estimator.
Trial Set 1.1: True Primary Equations Coefficients Comparison
Here I display the coefficent vector $\beta_1$ used to generate the data set corresponding to the position its file name appears in 'input_filenames0' above.
true_bcoeffs[0][0]
Trial Set 1.1: Primary Function Coefficient Estimates
Here I show the sampling distribution of the elements of $\hat{\beta}_1$.
psd.tbl_dsp(btables[0],1,5,line_nms[0],1)
psd.coeffden(bcoeff[0],line_nms[0],[-0.4,0.4],9,2,1,0,0)
psd.coeffden(bcoeff[0],line_nms[0],[-0.4,0.4],9,2,0,0,0)
Trial Set 1.1: Comments on Primary Function Coefficient Estimates
- Oracle is clearly the best estimator, followed by the known subset, lasso estimator, and unknown subset estimator. In both cases the bias of the lasso estimator more or less splits the difference between the the known and unknown estimator.
Trial Set 1.2: Estimator Comparison when $t_{inst} = 30$
Here we examine the sampling distribution of $\hat{\beta}_1, \hat{\alpha}_{1}$
- Number of Cross Sections: 10
- Number of Time Periods: 50
- Number of Endogenous Regressors: 1
- Number of Exogenous Regressors: 1
- Total Number of Instruments: 30
- Number of Instrument Relevant to Each Cross Section: 3
Trial Set 1.2: Merged DGP and Estimator Function Input Dictionary Comparison
Here I have merged together the dictionaries used to generate both the underlying dataset and the results (you will see the file name for this data set below) and the dictionary used to produce the estimates based on that data below. For a description of each variable name please see Variable Description
merged_dcts[1][0]
Trial Set 1.2: True Secondary Equation Coefficients Comparison
Here I display the coefficent vectors $\alpha_{1jd}$ used to generate the data set (by row indicating cross section and equation) corresponding to the position its file name appears in 'input_filenames0' above.
Note:
1.) The density of the secondary regression coefficient matrix is 13%
display_html(true_acoeffs[1][0][0])
Trial Set 1.2: Secondary Function Coefficient Estimates
Here I show characteristics of the sampling distributions of the elements of $\hat{\alpha}_{dj}$.
for i in range(1,len(atables[1])):
display(atables[1][i][0])
display(line_nms[1][i])
Trial Set 1.2: Comments on Secondary Function Coefficient Estimates
- Same comments roughly as 1.1
Trial Set 1.2: True Primary Equations Coefficients Comparison
Here I display the coefficent vector $\beta_1$ used to generate the data set corresponding to the position its file name appears in 'input_filenames0' above.
true_bcoeffs[1][0]
Trial Set 1.2: Primary Function Coefficient Estimates
Here I show the sampling distribution of the elements of $\hat{\beta}_1$.
psd.tbl_dsp(btables[1],1,5,line_nms[1],1)
psd.coeffden(bcoeff[1],line_nms[1],[-0.4,0.4],9,2,1,0,0)
psd.coeffden(bcoeff[1],line_nms[1],[-0.4,0.4],9,2,0,0,0)
Trial Set 1.2: Comments on Primary Function Coefficient Estimates
- The results here are similar to those in 1.1 but here there is a more clear seperation between the sampling distribution of the oracle an the rest.
Trial Set 1.3: Description
Here we examine the sampling distribution of $\hat{\beta}_1, \hat{\alpha}_{1}$.
- Number of Cross Sections: 10
- Number of Endogenous Regressors: 1
- Number of Exogenous Regressors: 1
- Number of Time Periods: 50
- Total Number of Instruments: 45
- Number of Instrument Relevant to Each Cross Section: 3
Trial Set 1.3: Merged DGP and Estimator Function Input Dictionary Comparison
Here I have merged together the dictionaries used to generate both the underlying dataset and the results (you will see the file name for this data set below) and the dictionary used to produce the estimates based on that data below. For a description of each variable name please see Variable Description
merged_dcts[2][0]
Trial Set 1.3: True Secondary Equation Coefficients Comparison
Here I display the coefficent vectors $\alpha_{1jd}$ used to generate the data set (by row indicating cross section and equation) corresponding to the position its file name appears in 'input_filenames0' above.
Note:
1.) A zero coefficient in the following matrix means that the instrument it multiplies is not relevant to that cross section.
2.) The density of the secondary regression coefficient matrix is 2%
display_html(true_acoeffs[2][0][0])
Trial Set 1.3: Secondary Function Coefficient Estimates
Here I show characteristics of the sampling distributions of the elements of $\hat{\alpha}_{dj}$.
for i in range(1,len(atables[2])):
display(atables[2][i][0])
display(line_nms[2][i])
Trial Set 1.3: Comments on Secondary Function Coefficient Estimates
- Due to the shrinkage inherent in the operation of the lasso estimator the bias of the coefficients is substantial and in nearly half the cases growing. However the variances of each are shrinking as the number of time periods grows.
Trial Set 1.3: True Primary Equations Coefficients Comparison
Here I display the coefficent vector $\beta_1$ used to generate the data set corresponding to the position its file name appears in 'input_filenames0' above.
true_bcoeffs[2][0]
Trial Set 1.3: Primary Function Coefficient Estimates
Here I show the sampling distribution of the elements of $\hat{\beta}_1$.
psd.tbl_dsp(btables[2],1,5,line_nms[2],1)
psd.coeffden(bcoeff[2],line_nms[2],[-0.4,0.4],9,2,1,0,0)
psd.coeffden(bcoeff[2],line_nms[2],[-0.4,0.4],9,2,0,0,0)
Trial Set 1.3: Comments on Primary Function Coefficient Estimates
- The behavior here is the same as the the known subset estimation in trials set 1.1
Trial Set 1.4: Lasso Comparison where $t_{inst} \in \{15,30,45,100\}$
Here we examine the sampling distribution of $\hat{\beta}_1, \hat{\alpha}_{1}$.
- Number of Cross Sections: 10
- Number of Endogenous Regressors: 1
- Number of Exogenous Regressors: 1
- Number of time periods: 50
- Number of Instrument Relevant to Each Cross Section: 3
Trial Set 1.4: Merged DGP and Estimator Function Input Dictionary Comparison
Here I have merged together the dictionaries used to generate both the underlying dataset and the results (you will see the file name for this data set below) and the dictionary used to produce the estimates based on that data below.
Below you will see a slider which can be used to summarize this merged dictionary corresponding to the position its file name appears in 'input_filenames0' above.
In accordance with the trial description, the only differences that should exist are the number of time periods (ntp) and the file name of the data set uded to generate the results.
Here I only display the dictionary for the new case where t_inst = 100
merged_dcts[3][3]
Trial Set 1.4: True Secondary Equation Coefficients Comparison
Here I display the coefficent vectors $\alpha_{1jd}$ used to generate the data set (by row indicating cross section and equation) corresponding to the position its file name appears in 'input_filenames0' above. Here they should also be identical across data sets.
Note:
1.) A zero coefficient in the following matrix means that the instrument it multiplies is not relevant to that cross section.
2.) In accordance with the description above they should be identical across results data sets.
3.) Here I only show the parameter matrix for the new case where there are 100 total instruments
display_html(true_acoeffs[3][3][0])
Trial Set 1.4: Secondary Function Coefficient Estimates
Here I interactively show the sampling distribution of the elements of $\hat{\alpha}_{dj}$.
for i in range(1,len(atables[3])):
display(atables[3][i][0])
display(line_nms[3][i])
Trial Set 1.4: Comments on Secondary Function Coefficient Estimates
- Due to the shrinkage inherent in the operation of the lasso estimator the bias of the coefficients is substantial and in nearly half the cases growing. However the variances of each are shrinking as the number of time periods grows.
Trial Set 1.4: True Primary Equations Coefficients Comparison
Here I interactively display the coefficent vector $\beta_1$ used to generate the data set corresponding to the position its file name appears in 'input_filenames0' above. Here they should be identical.
true_bcoeffs[3][0]
Trial Set 1.4: Primary Function Coefficient Estimates
Here I show the sampling distribution of the elements of $\hat{\beta}_1$.
psd.tbl_dsp(btables[3],1,5,line_nms[3],1)
psd.coeffden(bcoeff[3],line_nms[3],[-0.4,0.4],9,2,1,0,0)
psd.coeffden(bcoeff[3],line_nms[3],[-0.4,0.4],9,2,0,0,0)
Trial Set 1.4: Comments on Primary Function Coefficient Estimates
- One can cleary see a gradual increase in the bias and variance of both coefficients as the total number of instruments increases, this is what one would expect.
Trial Set 2.0: Properties of Lasso Estimator, Increasing Number of Cross Sections
Here we examine the sampling distribution of $\hat{\beta}_1$, and $\hat{\alpha}_{1}$ as the number of cross section increases.
- Number of Cross Sections: 10,25,40
- Number of Time Periods: 50
- Number of Endogenous Regressors: 1
- Number of Exogenous Regressors: 1
- Total Number of Instruments: 100
- Number of Instrument Relevant to Each Cross Section: 3
Trial Set 2.0: Merged DGP and Estimator Function Input Dictionary Comparison
Here I have merged together the dictionaries used to generate both the underlying dataset and the results (you will see the file name for this data set below) and the dictionary used to produce the estimates based on that data below. For a description of each variable name please see Variable Description
merged_dcts[4][0]
Trial Set 2.0: True Secondary Equation Coefficients Comparison
Here I display the coefficent vectors $\alpha_{1jd}$ used to generate the data set (by row indicating cross section and equation) corresponding to the position its file name appears in 'input_filenames3' above.
Note:
1.) A zero coefficient in the following matrix means that the instrument it multiplies is not relevant to that cross section.
2.) For the sake of presentation I only show the true parameter matrix for the case where ncs = 25
# psd.indict_dsp(true_acoeffs[4],2)
display_html(true_acoeffs[4][1][0])
Trial Set 2.0: Secondary Function Coefficient Estimates
Here I interactively show the sampling distribution of the elements of $\hat{\alpha}_{dj}$.
for i in range(1,len(atables[4])):
display(atables[4][i][0])
display(line_nms[4][i])
Trial Set 2.0: Comments on Secondary Function Coefficient Estimates
- Due to the lack of stability in lasso estiamtes its difficult to interpret
Trial Set 2.0: True Primary Equations Coefficients Comparison
Here I interactively display the coefficent vector $\beta_1$ used to generate the data set corresponding to the position its file name appears in 'input_filenames3' above. Here they should be identical.
true_bcoeffs[4][0]
Trial Set 2.0: Primary Function Coefficient Estimates
Here I show the sampling distribution of the elements of $\hat{\beta}_1$.
psd.tbl_dsp(btables[4],1,5,line_nms[4],1)
psd.coeffden(bcoeff[4],line_nms[4],[-0.4,0.4],9,2,1,0,0)
psd.coeffden(bcoeff[4],line_nms[4],[-0.4,0.4],9,2,0,0,0)
Trial Set 2.0: Comments on Primary Function Coefficient Estimates
- This is also what we would expect from a consistent estimator, a decrease in both bias an variance of the sampling distribution as the number of cross sections increases.
Trial Set 3.0: Properties of Lasso Estimator, Increasing Number of Time Periods
Here we examine the sampling distribution of $\hat{\beta}_1, \hat{\alpha}_{1}$.
- Number of Cross Sections: 10
- Number of Time Periods: 50,100,150
- Number of Endogenous Regressors: 1
- Number of Exogenous Regressors: 1
- Total Number of Instruments: 100
- Number of Instrument Relevant to Each Cross Section: 3
Trial Set 3.0: Merged DGP and Estimator Function Input Dictionary Comparison
Here I have merged together the dictionaries used to generate both the underlying dataset and the results (you will see the file name for this data set below) and the dictionary used to produce the estimates based on that data below. For a description of each variable name please see Variable Description
merged_dcts[5][0]
Trial Set 3.0: True Secondary Equation Coefficients Comparison
Here I display the coefficent vectors $\alpha_{1jd}$ used to generate the data set (by row indicating cross section and equation) corresponding to the position its file name appears in 'input_filenames0' above.
Note:
1.) A zero coefficient in the following matrix means that the instrument it multiplies is not relevant to that cross section.
display_html(true_acoeffs[5][0][0])
Trial Set 3.0: Secondary Function Coefficient Estimates
Here I show characteristics of the sampling distributions of the elements of $\hat{\alpha}_{dj}$, only for the new cases where ntp = 100,150
# display(psd.cfs_dsp(acoeff[5],atables[5],2,7,line_nms[5]))
for i in range(1,len(atables[5])):
display(atables[5][i][0])
display(line_nms[5][i])
Trial Set 3.0: Comments on Secondary Function Coefficient Estimates
- We can see and decrease in both bias and sampling variance as ntp increases.
Trial Set 3.0: True Primary Equations Coefficients Comparison
Here I display the coefficent vector $\beta_1$ used to generate the data set corresponding to the position its file name appears in 'input_filenames4' above.
true_bcoeffs[5][0]
Trial Set 3.0: Primary Function Coefficient Estimates
Here I show the sampling distribution of the elements of $\hat{\beta}_1$.
psd.tbl_dsp(btables[5],1,5,line_nms[5],1)
psd.coeffden(bcoeff[5],line_nms[5],[-0.4,0.4],9,2,1,0,0)
psd.coeffden(bcoeff[5],line_nms[5],[-0.4,0.4],9,2,0,0,0)
Trial Set 3.0: Comments on Primary Function Coefficient Estimates
- Precisely what we woud expect from a consistent and asymptotically normally distributed estimator.