Topics in econometrics: Identification, reproducibility, and education

What the talk is about

• I want to give an overview of some of my research interests and hope to get you interested in them too.

• Three themes:

  • Education
  • Identification
  • Reproducibility

• Education: Get all of us on the same page about the method of moments.

  • Why? Because I want you to know more about linear dynamic panel data methods.
  • What’s the point if software already exists? Many things we do not know very well.

• Identification: Can we uniquely learn an unknown parameter?

  • Why? Because we need to be able to communicate about the same thing.
  • What’s the point if we only look at unrealistic models? Hopefully they shed light on more complex models.

• Reproducibility: Trust, but verify.

  • Why? Because research resources are scarce.
  • What’s the point if it is too much work? Introduce a “spellchecker” instead.

Takeaway messages

• It is getting harder to pile on prerequisites to get to the point where a student is ready for advanced econometric methods. Emphasize a “shortest path” approach and exploit parallels across different toy models as much as possible.
• Even a linear AR(1) panel data model is a complicated toy model.
• If there is uncertainty about a report with respect to the instruments used, an algebraic check may be applied as a quick “spell-check”.

Outline

• Method of moments through examples
• Understand a popular identification argument used in linear dynamic panel data models
• Illustrate through a simulation of a toy model the (not very well-known) identification issues
• Show you reproducibility issues in a “simpler” IV setting

The method of moments

• You have been using the method of moments without you knowing it:

  • Sample averages, sample proportions
  • Least squares coefficients à la output from a Stata command like reg y x1 x2
  • Average treatment effects: difference of two means when applied to RCTs
  • Instrumental variables

• So why should you know more about the method?

• Because it is a powerful black box.

  • Many moving components you may not be aware of, even in “simple” settings
  • Generalized version of the method of moments
  • Affects the way we approach scientific pursuits
  • Affects the way we do consulting in industrial and policy settings

Making predictions

• One of the four words in the sentence “I SEE THE MOUSE” will be selected at random by a person. That person will tell you how many \(E\)’s there are in the selected word.

• Call \(Y\) the number of letters in the selected word and \(X_1\) the number of \(E\)’s in the selected word.

• Your task is to predict the number of letters in the selected word (ex ante).

• You would be “punished” according to the square of the difference between the actual \(Y\) and your prediction.

• Because you do not know for sure which word will be chosen, you have to allow for contingencies.

• Therefore, losses depend on which word was chosen.

• What would be your prediction rule in order to make your expected loss as small as possible?

• One way to answer this question is to propose a prediction rule and hope for the best.

• For example, we can say that the rule should have the form \(\beta_0+\beta_1 X_1\).

• The task is now to find the unique solution to the following optimization problem: \[\min_{\beta_{0},\beta_{1}}\mathbb{E}\left[\left(Y-\beta_{0}-\beta_{1}X_{1}\right)^{2}\right].\]

• An optimal solution \(\left(\beta_{0}^{*},\beta_{1}^{*}\right)\) solves the following first-order conditions: \[\begin{eqnarray*} \mathbb{E}\left(Y-\beta_{0}^{*}-\beta_{1}^{*}X_{1}\right) &=& 0, \\ \mathbb{E}\left(X_{1}\left(Y-\beta_{0}^{*}-\beta_{1}^{*}X_{1}\right)\right) &=& 0. \end{eqnarray*}\]

• As a result, we have \[\begin{equation}\beta_{0}^{*} = \mathbb{E}\left(Y\right)-\beta_{1}^{*}\mathbb{E}\left(X_{1}\right),\qquad\beta_{1}^{*}=\dfrac{\mathsf{Cov}\left(X_{1},Y\right)}{\mathsf{Var}\left(X_{1}\right)}.\label{blp-coef}\end{equation}\]

Connections: “line of best fit”

• If you know the population distribution of \(\left(X_1,Y\right)\), then you can figure out exact values of \(\left(\beta_{0}^{*},\beta_{1}^{*}\right)\).
• Otherwise, you need to estimate \(\left(\beta_{0}^{*},\beta_{1}^{*}\right)\).
• If you have IID copies of \(\left(X_1,Y\right)\), then you are back to what you already know!

Connections: method of moments

• In effect, you have replaced the \(\mathbb{E}\left(\cdot\right)\) with \(\dfrac{1}{n}\displaystyle\sum_{i=1}^n \left(\cdot\right)\).
• You have unwittingly used the method of moments!!
• To use the method, you need moment conditions which restrict the behavior of some functions of both the data and the unknown parameters.
• Where do you find these? Go back to first-order conditions.

Connections: models

• It is definitely possible to think of a generative model to make sense of linear regression.
• Define the error from best linear prediction to be \(u=Y-\beta_0^*-\beta_1^*X_1\).
• The generative model now has a signal plus noise form \(Y=\beta_0^*+\beta_1^* X_1 +u\).
• The moment conditions can be rewritten as: \[\mathbb{E}\left(u\cdot 1\right)=0,\ \ \mathbb{E}\left(u\cdot X_1\right)=0.\]

Returning to I SEE THE MOUSE

• If you do the calculations, the best linear predictor is \(\beta_0^*+\beta_1^* X_1=2+X_1\). You might find this result strange, but this is what you get!

• I can show you through a simulation. Try 10 IID draws from the joint distribution of \(\left(X_1,Y\right)\):

coefs <- matrix(NA, nrow=10^4, ncol=2)
for(i in 1:10^4)
{
  source <- matrix(c(1,3,3,5,0,2,1,1), ncol=2)
  data <- source[sample(nrow(source),size=10,replace=TRUE),]
  temp <- lm(data[,1]~data[,2])
  coefs[i, ] <- coef(temp)
}

• Now try 100 IID draws.

• Some of you might have felt weird about the prediction. If you thought of something like:

  • If \(X=0\), then I predict \(Y\) to be 1.
  • If \(X=2\), then I predict \(Y\) to be 3.
  • If \(X=1\), then I predict \(Y\) to be either 3 or 5.

• Then, you have travelled unwittingly on a nonparametric path and are, perhaps without knowing, questioning the specification of the linear prediction rule!

Instrumental variables

• What if you are in a situation where \(Y=\beta_0+\beta_1X_1+u\) is not a linear regression?

• What this means:

  • \(\beta_0\) and \(\beta_1\) are not necessarily the coefficients of the best linear predictor of \(Y\) using \(X_1\).
    
  • \(\mathbb{E}\left(u \cdot 1\right) \neq 0\) or \(\mathbb{E}\left(u \cdot X_1\right) \neq 0\) or both

• But \(\beta_1\) here is still the causal effect of \(X_1\) on \(Y\).

• One approach is to find an external source of variation \(Z\) correlated with \(X_1\) but not correlated with \(u\).
• The argument is as follows: \[\mathsf{Cov}\left(Y, Z\right)=\beta_1\mathsf{Cov}\left(X_1, Z\right)+\mathsf{Cov}\left(u, Z\right)\] \[\Rightarrow\beta_1=\dfrac{\mathsf{Cov}\left(Y, Z\right)}{\mathsf{Cov}\left(X_1, Z\right)}.\]

• The previous identification argument requires

  • \(\mathsf{Cov}\left(X_1, Z\right)\neq 0\)
  • \(\mathsf{Cov}\left(u, Z\right)=0\)

• Violations of any of the two aforementioned conditions would already signal a failure of instrumental variables to identify \(\beta_1\).

• Observe that \[\beta_1=\dfrac{\mathsf{Cov}\left(Y, Z\right)}{\mathsf{Cov}\left(X_1, Z\right)}=\dfrac{\mathsf{Cov}\left(Y, X\right)}{\mathsf{Var}\left(X_1\right)}\left(\dfrac{\mathsf{Cov}\left(X_1, Z\right)}{\mathsf{Var}\left(X_1\right)}\right)^{-1}\]

• Therefore, \(\beta_1\) may be interpreted as a ratio of two regression slopes!

• Since there are actually two unknown parameters \(\beta_0\) and \(\beta_1\), we can write down two moment conditions to recover both using instrumental variables: \[ \mathbb{E}\left(u \cdot 1 \right)=0,\ \ \mathbb{E}\left(u \cdot Z\right) =0\] \[\Rightarrow \mathbb{E}\left(u \cdot 1 \right)=0,\ \ \mathsf{Cov}\left(u, Z\right) =0\] \[\Rightarrow \begin{eqnarray*} \mathbb{E}\left(\left(Y-\beta_0-\beta_1 X_1\right) \cdot 1 \right)=0 \\ \mathbb{E}\left(\left(Y-\beta_0-\beta_1 X_1\right) \cdot Z\right) =0\end{eqnarray*}\]

Outline

• Method of moments through examples
• Understand a popular identification argument used in linear dynamic panel data models
• Illustrate through a simulation of a toy model the identification issues
• Show you reproducibility issues in a “simpler” IV setting

A panel data setting

• Setup:

  • Units indexed by \(i\) (but I will suppress this notation)
  • \(t=1,2,3\) index the time periods
  • \(Y_t\), \(Y_{t-1}\), \(W\): current \(Y\), previous period’s \(Y\), extra information

• Task: Predict \(Y_t\) using \(Y_{t-1}\) and \(W\).
• This is just going to extend the best (linear) prediction example. Nothing special here!
• So, we are just in a linear regression setting: \[Y_t=\beta_0^*+\beta_1^* Y_{t-1}+\beta_2^* W+u_t\]

• More complicated task: Predict \(Y_t\) using \(Y_{t-1}\) and \(W\). But \(W\) is now unobserved.
• This task is much harder to achieve, but we can settle for predicting changes in \(Y\) instead.
• Calculate a differenced equation: \[Y_3-Y_2 = \beta_1^* \left(Y_2-Y_1\right)+\left(u_3-u_2\right)\]
• New wrinkle: Not a linear regression anymore!

• \(\beta_1^*\) has a predictive interpretation, but the differenced equation effectively becomes a structural equation.
• Here, we can use IV! Consider a potential instrumental variable where \(Z=Y_1\).
• Mimic an earlier argument: \[\begin{eqnarray*} \mathrm{Cov}\left(\Delta Y_{3},Y_1\right) & = & \mathrm{Cov}\left(\beta_{1}^*\Delta Y_{2}+\Delta u_{3},Y_1\right)\\ \mathrm{Cov}\left(\Delta Y_{3},Y_1\right) & = & \beta_{1}^*\mathrm{Cov}\left(\Delta Y_{2},Y_1\right)+\mathrm{Cov}\left(\Delta u_{3},Y_1\right)\\ \dfrac{\mathrm{Cov}\left(\Delta Y_{3},Y_1\right)}{\mathrm{Cov}\left(\Delta Y_{2},Y_1\right)} & = & \beta_{1}^*. \end{eqnarray*}\]

When will the previous argument be plausible?

• Need \(\mathrm{Cov}\left(\Delta Y_{2},Y_1\right)\neq 0\) and \(\mathrm{Cov}\left(\Delta u_{3},Y_1\right)=0\)

• Note that \(Y_t\) is nothing but an accummulation of past \(u_t\)’s plus an initial starting point \(Y_1\).

• But there are ways to guarantee \(\mathrm{Cov}\left(\Delta u_{3},Y_1\right)=0\):

  • Posit that \(u_t\) is a “shock”: Show evidence that \(\mathbb{E}\left(u_t|Y_{t-1},...\right)=0\) for all \(t\).
  • Posit that \(u_t\) does not have serial correlation over time.

• Guaranteeing \(\mathrm{Cov}\left(\Delta Y_{2},Y_1\right)\neq 0\) is slightly tricky.

What if there is another observed time period?

• We can also write down the moment condition used to identify \(\beta_1^*\): \[ \mathbb{E}\left(\left(\Delta Y_3-\beta_1^* \Delta Y_2\right) \cdot Y_1\right) =0\]

• What happens if we observe \(Y_4\)? Then you will have extra moment conditions (of course, there is a price!) because you can take other differences.

• Again, you observe \(\left(Y_1,Y_2,Y_3,Y_4\right)\).

• You have something like: \[ \begin{eqnarray*} \mathbb{E}\left(\left(\Delta Y_3-\beta_1^* \Delta Y_2\right) \cdot Y_1\right) =0\\ \mathbb{E}\left(\left(\Delta Y_4-\beta_1^* \Delta Y_3\right) \cdot Y_1\right) =0 \\ \mathbb{E}\left(\left(\Delta Y_4-\beta_1^* \Delta Y_3\right) \cdot Y_2\right) =0\end{eqnarray*}\]

• In the end, you will have more moment conditions than the dimension of the parameter \(\beta_1^*\).

• How do you combine them? This is where the generalized method of moments (GMM) comes in.

• But a particularly important moment condition requires four observed time periods to work.

  • This nonlinear (NL) moment condition is based on Ahn and Schmidt (1995): \[ \mathbb{E}\left( \left(Y_4-\beta_1^* Y_3\right) \cdot \left(\Delta Y_3-\beta_1^* \Delta Y_2\right)\right) =0\]
  • Nobody has used NL in empirical work, with only about 1 exception since 1995!

• There is another important moment condition which requires an extra assumption called mean stationarity for the moment condition to be valid.

Outline

• Method of moments through examples
• Understand a popular identification argument used in linear dynamic panel data models
• Illustrate through a simulation of a toy model the identification issues
• Show you reproducibility issues in a “simpler” IV setting

A class of DGPs to show that linear AR(1) panel data model is complicated

• I will illustrate using Monte Carlo simulation.

• The model you have seen so far is not enough to allow you to generate artificial data.

• I need to specify more details in order to generate data from a linear AR(1) panel data model.

• Set four time periods to observe \[Y_t=\underbrace{\beta_0^*+ \beta_2^*W}_{c}+\beta_1^* Y_{t-1} +u_t\]
• More explicitly: \[\begin{eqnarray*}Y_2 =c+\beta_1^* Y_1+u_2 \\ Y_3 =c+\beta_1^* Y_2+u_2 \\ Y_4 = c+\beta_1^* Y_3+u_4 \end{eqnarray*}\]

• IID across \(i\), so I suppressed this notation
• Initialization: \[\left(\begin{array}{c} c\\ Y_1 \end{array}\right)\sim N\left(\left(\begin{array}{c} 0 \\ 0 \end{array}\right),\left(\begin{array}{cc}\sigma_{c}^{2} & \mathsf{Cov}\left(c,Y_1\right)\\\mathsf{Cov}\left(c,Y_1\right) & \mathsf{Var}\left(Y_1\right)\end{array}\right)\right)\]
• Errors: \[\left(u_{2},u_{3}, u_{4}\right) \sim N\left(0, \mathsf{diag}\left(\sigma^2_2,\sigma^2_3, \sigma^2_4\right)\right)\]

• We will look at four data generating processes (DGPs) or “generative models” representing different parameter configurations.
• I will focus on NL alone to give you a sense of what is happening.
• Then I will make a small comparison between NL and the Arellano-Bond (AB) GMM estimator for \(\beta_1^*\) which exploits the three moment conditions earlier.

DGP1

DGP1, increase \(n\) tenfold

DGP2

DGP2, increase \(n\) tenfold

DGP3

DGP4

Is this a big deal?

Take \(n=50\):

Now, \(n=800\):

Outline

• Method of moments through examples
• Understand a popular identification argument used in linear dynamic panel data models
• Illustrate through a simulation of a toy model the identification issues
• Show you reproducibility issues in a “simpler” IV setting

The paper being reproduced (BC)

BC’s main specification

\[\begin{eqnarray*}\mathrm{budget share}_{iht} &=& \alpha_i + \beta_i \log \mathrm{expenditure}_{iht} \\ && +\gamma_i\mathrm{budget share}_{ih,t-1} + \sum_k \delta_{ik}\mathrm{controls}_{kht} \\ && +\underbrace{\rho_{ih}+\varepsilon_{iht}}_{u_{iht}}\end{eqnarray*}\]

• \(i\) indexes goods in consumption basket, \(h\) indexes households, \(t\) indexes quarter of survey
• sign and magnitude of \(\gamma_i\), magnitude of \(\beta_i/(1-\gamma_i)\)

What challenges GMM bring to reproducibility

• Many decisions go into a even bigger black box.

  • The number and type of moment conditions
  • The weighting matrix
  • How covariates and/or dummy variables are included and treated
  • Iteration and possible bias correction of estimation procedure

BC’s identifying assumptions: formula display

\[\begin{eqnarray*} &\mathbb{E}\left(\Delta\varepsilon_{iht} \log \mathrm{expenditure}_{h,t-k}\right) = 0, \ \ \forall k\geq 1 \\ &\mathbb{E}\left(\Delta\varepsilon_{iht} \log \mathrm{income}_{h,t-k}\right) = 0, \ \ \forall k \\ &\mathbb{E}\left(u_{iht} \Delta\log \mathrm{expenditure}_{h,t-k}\right) = 0, \ \ \forall k\geq 1 \\ &\mathbb{E}\left(u_{iht} \Delta\log \mathrm{income}_{h,t-k}\right) = 0, \ \ \forall k \end{eqnarray*}\]

What makes BC special

• Bespoke application of linear dynamic panel data methods

  • Moment conditions are based on external instruments instead of “internal” instruments
  • Used iterated GMM and applies weak instrument tests specifically for GMM

• Spanish panel has a rotating panel structure

• Uses microdata as a sanity check for macro models

• But we were unable to reproduce the results, as there is uncertainty about the instrument set.

Lower the marginal cost of reproducibility

• Perhaps the standard is not to reproduce published results exactly.
• Reduce “hard” feelings, lower defensiveness, reduce human resources…
• We now show you a more minimal standard which gives the maximum benefit of the doubt to the published results.
• Idea is to exploit algebraic relations arising from the uncertainty about the instrument set.

Illustrating with a toy example: ingredients from report

• Data from Angrist and Krueger (1991)
• Model: \[\log\left(\mathrm{WKLYWGE}\right)=\beta_0+\beta_1 \mathrm{EDUC}+u\]
• Outcome \(Y\): log weekly wages
• Regressors \(X\): 1 and years of schooling

• Reported instruments \(Z\): Age and age squared
• Actual instruments used: Age and age squared
• Reported focus coefficient \(\psi^\prime \widehat{\theta}_R=\widehat{\beta}_1\): returns to schooling
• Reported SE: \(\left(\mathsf{Var}\left(\psi^\prime\widehat{\theta}_R\right)\right)^{1/2}\)

actual <- ivreg(LWKLYWGE ~ EDUC | AGEQ + AGEQSQ, data=ak2029)
# Reported focus coefficient and SE
c(coef(actual)["EDUC"], sqrt(vcov(actual)["EDUC", "EDUC"]))

       EDUC             
0.057426344 0.008433067 

Illustrating with a toy example: ingredients from the checker

• Split reported instrument set into \(Z=\left[\begin{array}{cc} Z_1 & Z_{2}\end{array}\right]\) with \(Z_2\) is the matrix of excluded instruments
• Augment the reported model: \[\log\left(\mathrm{WKLYWGE}\right)=\beta_0+\beta_1 \mathrm{EDUC}+\gamma^\prime Z_2+u\]
• Estimate the augmented model by IV with \(Z\) as the instrument set.

• Obtain the estimate of the focus parameter \(\psi^\prime \widehat{\theta}=\widetilde{\beta}_1\) and its associated standard error \(\left(\mathsf{Var}\left(\psi^\prime\widehat{\theta}\right)\right)^{1/2}\).
• Calculate coefficient contrast \(\psi^\prime\widehat{\theta}-\psi^\prime\widehat{\theta}_R\) and variance contrast \(\mathsf{Var}\left(\psi^\prime \widehat{\theta}\right)-\mathsf{Var}\left(\psi^\prime\widehat{\theta}_R\right)\).

# Unrestricted model
just1 <- ivreg(LWKLYWGE ~ EDUC +AGEQ | AGEQ + AGEQSQ, data=ak2029)
# Coefficient contrast
coef(just1)["EDUC"]-coef(actual)["EDUC"]

     EDUC 
0.4452792 

# Variance contrast must be positive
vcov(just1)["EDUC", "EDUC"]-vcov(actual)["EDUC", "EDUC"]

[1] 0.05058544

• Obtain the Wald statistic for testing the null \(\gamma=0\).

library(car)
linearHypothesis(just1, "AGEQ", test="Chisq")

Linear hypothesis test

Hypothesis:
AGEQ = 0

Model 1: restricted model
Model 2: LWKLYWGE ~ EDUC + AGEQ | AGEQ + AGEQSQ

  Res.Df Df  Chisq Pr(>Chisq)  
1 247197                       
2 247196  1 3.9508    0.04685 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Illustrating with a toy example: bounds

Calculate the bounds:

\[\psi^\prime \widehat{\theta} - \frac{1}{2}\left(\cos 2\alpha \pm 1\right)\left(\mathsf{Var}\left(\psi^\prime \widehat{\theta}\right)-\mathsf{Var}\left(\psi^\prime\widehat{\theta}_R\right)\right)^{1/2} \\ \times \left(\mathrm{Wald \ stat}\right)^{1/2}\]

where \(\cos 2\alpha\) is given by

\[\frac{\psi^\prime\widehat{\theta}-\psi^\prime\widehat{\theta}_R}{\left(\mathsf{Var}\left(\psi^\prime \widehat{\theta}\right)-\mathsf{Var}\left(\psi^\prime\widehat{\theta}_R\right)\right)^{1/2}\left(\mathrm{Wald \ stat}\right)^{1/2}}\]

• If reported estimate of the focus parameter is not in the bounds, then something is wrong with the instrument list.
• Otherwise, we can tentatively accept the reported results even if we cannot reproduce the results exactly.

# cos 2\alpha must be between -1 and 1
0.445/(sqrt(0.051)*sqrt(26.13))

[1] 0.385483

# Extreme bounds
c(0.503-0.5*(0.445/(sqrt(0.051)*sqrt(3.951))+1)*(sqrt(0.051)*sqrt(3.951)), 0.503-0.5*(0.445/(sqrt(0.051)*sqrt(3.951))-1)*(sqrt(0.051)*sqrt(3.951)))

[1] 0.05605569 0.50494431

• Reported free coefficient is 0.057 and it is inside these bounds.

Illustrating with a toy example: another check

• Repeat the same exercise but this time the excluded instrument is AGEQSQ.

# cos 2\alpha must be between -1 and 1
0.634/(sqrt(0.198)*sqrt(2.045))

[1] 0.9963456

# Extreme bounds
c(0.692-0.5*(0.634/(sqrt(0.198)*sqrt(2.045))+1)*(sqrt(0.198)*sqrt(2.045)), 0.692-0.5*(0.634/(sqrt(0.198)*sqrt(2.045))-1)*(sqrt(0.198)*sqrt(2.045)))

[1] 0.05683731 0.69316269

• Reported free coefficient is 0.057 and it is inside these bounds.

Illustrating with a toy example: something is wrong

• Reported instruments used: Age and marital status
• Actual instruments used: Age and age squared
• Actual instruments used are unknown to us in reality, but we rely on the report.

The only ingredients from the publication side

• The cleaned data
• The list of instruments or moment conditions used
• The estimate of the key or focus coefficients
• The corresponding standard errors and the details which led to the reported standard errors
• Bottom line: all these are in the published report!

Sketch of the idea: notation

• \(X\) regressor matrix, some exogenous and some endogenous
• \(Z=\left[\begin{array}{cc} Z_1 & Z_{2}\end{array}\right]\) with \(Z_2\) is the matrix of excluded instruments
• \(S=\left[\begin{array}{cc} X & Z_{2}\end{array}\right]\)
• \(\theta=\left[\begin{array}{cc} \beta^{\prime} & \gamma^{\prime}\end{array}\right]^{\prime}\)
• \(\widehat{W}\) weighting matrix
• \(\psi^\prime\theta\) is the focus parameter, typically an element of \(\beta\)
• Prior linear constraints as \(C\theta=c\), \(C\) has full row rank

Sketch of the idea: setup

• Think about IV regression of \(Y\) on \(S=\left[\begin{array}{cc} X & Z_{2}\end{array}\right]\).
• Let \(M\) be an arbitrary matrix of full row rank. Find the range of estimates of \(\psi^{\prime}\theta\), when imposing linear constraints of the form \(M\left(C\theta-c\right)=0\).
• From a reproducibility standpoint, we are uncertain about the instrument list (hence \(Z\)) disclosed in the published report.
• Therefore, the case where \(C=\left(\begin{array}{cc} 0 & I\end{array}\right)\) and \(c=0\) is the relevant configuration. Hence, \(\gamma=0\).

Sketch of the idea: algebra

• Unconstrained optimization problem: \[\widehat{\theta}=\arg\min_{\theta}\left(Y-S\theta\right)^{\prime}Z\widehat{W}^{-1}Z^{\prime}\left(Y-S\theta\right)\]
• Constrained optimization problem: \[\widehat{\theta}_R=\arg\min_{\theta:\ M\left(C\theta-c\right)=0 }\left(Y-S\theta\right)^{\prime}Z\widehat{W}^{-1}Z^{\prime}\left(Y-S\theta\right)\]

• There is an algebraic relationship between \(\psi^\prime \widehat{\theta}\) and \(\psi^\prime \widehat{\theta}_R\).
• Sweeping through all possible \(M\) leads to extreme bounds.

Wrap up

• Illustrating how to get students up to speed to advanced topics
• Linear AR(1) panel data models are more complicated than they appear
• Extreme bounds idea comes from Leamer (1983), but extended to IV case, repurposed for reproducibility checks
• Many loose ends to tie up and hope they pique your interest