slide-weak-instruments.knit

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# Advanced Econometrics III

## (高级计量经济学III 全英文)

### Hu Huaping (胡华平 )

### NWAFU (西北农林科技大学)

### School of Economics and Management (经济管理学院)

### huhuaping01 at hotmail.com

### 2025-04-28

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name: session-weak-iv

# Session (Handeling Weak Instruments)

[Lots of Instrumental Variables](#lots-of-iv)

[Weak Instruments](#weak-instruments)

---
layout: false
class: center, middle, duke-softblue,hide_logo
name: lots-of-iv

## Lots of Instrumental Variables

---
layout: true
 
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<div class="my-footer">huhuaping@ <a href="#chapter17"> Session: Weak Instumental Variables |</a>
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="#definition"> Lots of Instrumental Variables </a> </div>

---

### Introduction

> "In instrumental variables regression, the instruments are called weak if their correlation with the endogenous regressors, conditional on any controls, is close to zero."

> --- Andrews,StockandSun (2018)
---

### Weak instruments

- Whereas exclusion restriction is not testable, the non-zero first stage
- Weak instruments can happen if the two variables are independent or the sample is small
- If you have a weak instrument, then the bias of 2SLS is centered on the bias of OLS and the cure ends up being worse than the disease
- This brought into sharp focus with Angrist and Krueger (1991) quarter of birth study and some papers that followed

---

### March 2022 Interview with Angrist

Before we dive into the paper, though, let's listen to Angrist discuss the history
https://youtu.be/ApNtXe-JDfA?t=2348
Somewhat inspiring to hear how Angrist reframed the weak instrument problem which his paper with Krueger brought into crisp focus

---

### Angrist and Krueger (1991)

- In practice, it is often difficult to find convincing instruments usually because potential instruments don't satisfy the exclusion restriction
- But in an early paper in the causal inference movement, Angrist and Krueger (1991) wrote a very interesting and influential study instrumental variable
- They were interested in schooling's effect on earnings and instrumented for it with which quarter of the year you were born
- Remember "strangeness principle"

> - why would birth quarter cause earnings in the reduced form (First stage of TSLS)?

---

### Compulsory schooling

- In the US, you could drop out of school once you turned 16
- "School districts typically require a student to have turned age six by January 1 of the year in which he or she enters school" (Angrist and Krueger 1991, p. 980)
- Children have different ages when they start school, though, and this creates different lengths of schooling at the time they turn 16 (potential drop out age):

If you're born in the fourth quarter, you hit 16 with more schooling than those born in the first quarter

---

### Visuals

- You need good data visualization for IV partly because of the scrutiny around the design
- The two pieces you should be ready to build pictures for are the first stage and the reduced form
- Angrist and Krueger (1991) provide simple, classic and compelling pictures of both

---

### Visuals: First Stage of TSLS

Men born earlier in the year have lower schooling. This indicates that there is a first stage.

Notice all the age cohorts 30s and 40s at the bottom. But then notice how it attenuates over time ...

.pull-left[

]

.pull-right[

<img src="https://cdn.mathpix.com/cropped/2025_04_24_8b8d91658673077f1454g-06.jpg?height=742&width=1060&top_left_y=1083&top_left_x=139" style="display: block; margin: auto;" />
]

---

### Visuals: Fisrt Stage of TSLS

Do differences in schooling due to different quarter of birth translate into different earnings?

---
exclude: true

### Two Stage Least Squares model (Setup)

- The causal model is

$$
`\begin{aligned}
Y_{i} &= X \pi+\delta S_{i}+\varepsilon \\
\end{aligned}`
$$

- The first stage regression is:

$$
`\begin{aligned}
S_{i} &= X \pi_{10}+\pi_{11} Z_{i}+\eta_{1 i} \\
\end{aligned}`
$$

- The reduced form regression is:

$$
`\begin{aligned}
Y_{i} &= X \pi_{20}+\pi_{21} Z_{i}+\eta_{2 i} \\
\end{aligned}`
$$

- The sample analog of the Wald estimator that adjusts for covariates:
`$\frac{\pi_{21}}{\pi_{11}}$`

---

### Two Stage Least Squares model (Instruments)

- Angrist and Krueger (1991) instrument for schooling using three quarter of birth dummies: a dummies for 1st, 2nd and 3rd `Qob`
- Their initial first-stage regression is:

$$
`\begin{aligned}
S_{i} &= X \pi_{10}+Z_{1 i} \pi_{11}+Z_{2 i} \pi_{12}+Z_{3 i} \pi_{13}+\eta_{1} \\
\end{aligned}`
$$

- The second stage is the same as before (including all controls `$X$` ), but the fitted values are from the new first stage

$$
`\begin{aligned}
Y_{i} &= X \pi+\delta \widehat{S}_{i}+\epsilon \\
\end{aligned}`
$$

---

### Data and variables summary (1/3)

The data set contains men from the official 1970/1980 Census, 5 percent Public Use Sample.

Sample for men born in:

- Born in 1920-1929 (1920s, age cohort 50-59): `$n = 247,199$`
- Born in 1930-1939 (1930s, age cohort 40-49): `$n = 329,509$`
- Born in 1940-1949 (1940s, age cohort 30-39): `$n = 486,926$`

---

### Data and variables summary (2/3)

Model's intercept:

- All models include an intercept term due to the dummy variables (not shown in the table).

Dependent variables:

- `Ln(W)`: Log of weekly earnings (mainly used)
- An MA(+2, -2) trend term was subtracted from each dependent variable. 
- `Ln(Wks)`: Log of weeks worked (for comparison only)

Interested independent variables:

- `S`: Total years of schooling
- two age variables (measured in quarters of years): `Age` and `Age^2`
- 50 residence state dummies: `State_1`, `State_2`, ..., `State_50`

.footnote[

Two age variables are used here to control for within-year-of-birth age effects on earnings.
]

---

### Data and variables summary (3/3)

Control variables included:

- `RACE` (1 = black)
- `SMSA` (1 = center city)
- `MARRIED` (1 = married)

Available instruments:

- 3 quarter-of-birth dummies: `Qob_1`, `Qob_2`, `Qob_3`
- 9 year-of-birth dummies: `YoB_1`, `YoB_2`, ..., `YoB_9`
- 8 region-of-residence dummies: `Reg_1`, `Reg_2`, ..., `Reg_8`
- 27 interaction terms between 3 quarter-of-birth dummies and 9 year-of-birth dummies
- 150 interaction terms between 3 quarter-of-birth dummies and 50 state dummies

---
exclude: true

### First stage regression (results)

.page-font-20[

| Outcome variable | Birth cohort | Mean | Quarter-of-birth effect `${ }^{\text {a }}$` | | | `$F$`-test `${ }^{\text {b }}$` [ `$P$`-value] |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| | | | I | II | III | |
| Total years of education | 1930-1939 | 12.79 | -0.124 (0.017) | -0.086 (0.017) | -0.015 (0.016) | 24.9 [0.0001] |
| | | | | | | |
| | 1940-1949 | 13.56 | -0.085 (0.012) | -0.035 (0.012) | -0.017 (0.011) | 18.6 [0.0001] |
| | | | | | | |
| High school graduate | 1930-1939 | 0.77 | -0.019 (0.002) | -0.020 (0.002) | -0.004 (0.002) | 46.4 [0.0001] |
| | | | | | | |
| | 1940-1949 | 0.86 | -0.015 (0.001) | -0.012 (0.001) | -0.002 (0.001) | 54.4 [0.0001] |
| | | | | | | |
| Years of educ. for high school graduates | 1930-1939 | 13.99 | -0.004 (0.014) | 0.051 (0.014) | 0.012 (0.014) | 5.9 [0.0006] |
| | | | | | | |
| | 1940-1949 | 14.28 | 0.005 (0.011) | 0.043 (0.011) | -0.003 (0.010) | 7.8 [0.0017] |
| | | | | | | |
| College graduate | 1930-1939 | 0.24 | -0.005 (0.002) | 0.003 (0.002) | 0.002 (0.002) | 5.0 [0.0021] |
| | | | | | | |
| | 1940-1949 | 0.30 | -0.003 (0.001) | 0.004 (0.001) | 0.000 (0.001) | 5.0 [0.0018] |

]

---

### Quarter of birth as a proxy for Schooling years

.pull-left[

]

.pull-right[

Quarter of birth (`Qob`) is a strong predictor of total years of education (`S`).

$$
`\begin{aligned}
S_{i} &=  \pi_{10}+ \pi_{11} Qob_1 + \pi_{12} Qob_2 \\
&+ \pi_{13} Qob_3 + v_{i} \\
\end{aligned}`
$$

- Standard errors are in parentheses.

- F-statistic is for a test of the hypothesis that the quarter-of-birth dummies jointly have no effect.

- F test all significant for all cohorts.

]

.footnote[
- **TABLE I**: The Effect of Quarter of Birth on Various Educational Outcome Variables
- **Source**: Angrist and Krueger (1991)
]

---
exclude: true

### IV Estimates Birth Cohorts 20-29, 1980 Census

| Independent variable | (1) OLS | (2) TSLS |
| :--- | :---: | :---: |
| Years of education | 0.0711 (0.0003) | 0.0891 (0.0161) |
| Race ( 1 = black) | - | - |
| SMSA (1 = center city) | - | - |
| Married (1 = married) | - | - |
| 9 Year-of-birth dummies | Yes | Yes |
| 8 Region-of-residence dummies | No | No |
| Age | - | - |
| Age-squared | - | - |
| `$\chi^{2}$` [dof] | - | 25.4 [29] |

---
exclude: true

### Second stage regression (results Table IV)

.page-font-20[

| Independent variable | (1) OLS | (2) TSLS | (3) OLS | (4) TSLS | (5) OLS | (6) TSLS | (7) OLS | (8) TSLS |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Years of education | 0.0802 (0.0004) | 0.0769 (0.0150) | 0.0802 (0.0004) | 0.1310 (0.0334) | 0.0701 (0.0004) | 0.0669 (0.0151) | 0.0701 (0.0004) | 0.1007 (0.0334) |
| Race (1 = black) | - | - | - | - | 0.2980 (0.0043) | -0.3055 (0.0353) | -0.2980 (0.0043) | -0.2271 (0.0776) |
| SMSA (1 = center city) | - | - | - | - | 0.1343 (0.0026) | 0.1362 (0.0092) | 0.1343 (0.0026) | 0.1163 (0.0198) |
| Married (1 = married) | - | - | - | - | 0.2928 (0.0037) | 0.2941 (0.0072) | 0.2928 (0.0037) | 0.2804 (0.0141) |
| 9 Year-of-birth dummies | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes |
| 8 Region of residence dummies | No | No | No | No | Yes | Yes | Yes | Yes |
| Age | - | - | 0.1446 (0.0676) | 0.1409 (0.0704) | - | - | 0.1162 (0.0652) | 0.1170 (0.0662) |
| Age-squared | - | - | -0.0015 (0.0007) | -0.0014 (0.0008) | - | - | -0.0013 (0.0007) | -0.0012 (0.0007) |
| `$\chi^{2}$` [dof] | - | 36.0 [29] | - | 25.6 [27] | - | 34.2 [29] | - | 28.8 [27] |

]

.footnote[
- **TABLE IV**: OLS and TSLS Estimates of the Return to Education for Men Born **1920-1929** : 1970 Census
- **Source**: Angrist and Krueger (1991)
]

---

### Excluded instruments (TABLE V, Age cohort 40-49)

AK1991 specify different excluded instruments in TSLS models in `TABLE V` ：

- 9 for column (2)
- 7 for column (3)
- 30 for column (4)
- 28 for column (5)

Other notes for `TABLE V`:

- Age and age-squared are measured in quarters of years.
- Each equation also includes an intercept term. 
- The sample size is 329,509.

---

### OLS VS TSLS Results (TABLE V, Age cohort 40-49)

.page-font-20[

| Independent variable | (1) OLS | (2) TSLS | (3) OLS | (4) TSLS | (5) OLS | (6) TSLS | (7) OLS | (8) TSLS |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Years of education | 0.0711 (0.0003) | 0.0891 (0.0161) | 0.0711 (0.0003) | 0.0760 (0.0290) | 0.0632 (0.0003) | 0.0806 (0.0164) | 0.0632 (0.0003) | 0.0600 (0.0299) |
| Race (1 = black) | - | - | - | - | -0.2575 (0.0040) | -0.2302 (0.0261) | -0.2575 (0.0040) | -0.2626 (0.0458) |
| SMSA (1 = center city) | - | - | - | - | 0.1763 (0.0029) | 0.1581 (0.0174) | 0.1763 (0.0029) | 0.1797 (0.0305) |
| Married (1 = married) | - | - | - | - | 0.2479 (0.0032) | 0.2440 (0.0049) | 0.2479 (0.0032) | 0.2486 (0.0073) |
| 9 Year-of-birth dummies | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes |
| 8 Region-of-residence dummies | No | No | No | No | Yes | Yes | Yes | Yes |
| Age | - | - | -0.0772 (0.0621) | -0.0801 (0.0645) | - | - | -0.0760 (0.0604) | -0.0741 (0.0626) |
| Age-squared | - | - | 0.0008 (0.0007) | 0.0008 (0.0007) | - | - | 0.0008 (0.0007) | 0.0007 (0.0007) |
| `$\chi^{2}$` [dof] | - | 25.4 [29] | - | 23.1 [27] | - | 22.5 [29] | - | 19.6 [27] |
| .red[Number of excluded instruments] | | 9 | | 7 | | 30 | | 28 |

]

.footnote[
- **TABLE V**: OLS and TSLS Estimates of the Return to Education for Men Born **1930-1939** : 1980 Census
- **Source**: Angrist and Krueger (1991)
]

---

### Model illustration: Column (5) OLS in Table V

The OLS model is:

$$
`\begin{aligned}
Ln(W_{i}) &= \delta_{0} +\delta_{1} S_{i} + \beta_{1} Race_i + \beta_{2} SMSA_i + \beta_{3} Married_i \\
&+ \beta_{4} YoB_1 + \beta_{5} YoB_2 + \beta_{6} YoB_3  + \beta_{7} YoB_4  + \beta_{8} YoB_5 \\
&+ \beta_{9} YoB_6 + \beta_{10} YoB_7 + \beta_{11} YoB_8 + \beta_{12} YoB_9 \\
&+ \beta_{13} Reg_1 + \beta_{14} Reg_2 + \beta_{15} Reg_3 + \beta_{16} Reg_4 + \beta_{17} Reg_5 \\
&+ \beta_{18} Reg_6 + \beta_{19} Reg_7 + \beta_{20} Reg_8 \\
&+ \epsilon_{i} \\
\end{aligned}`
$$

---

### Model illustration: Column (6) TSLS in Table V

.page-font-20[

- First stage (reduced equation):

$$
`\begin{aligned}
S_{i} &= \pi_{10} +\pi_{11} Race_i + \pi_{12} SMSA_i + \pi_{13} Married_i  \\
&+ \pi_{14} YoB_1 + \pi_{15} YoB_2 + \pi_{16} YoB_3  + \pi_{17} YoB_4  + \pi_{18} YoB_5 \\
&+ \pi_{19} YoB_6 + \pi_{110} YoB_7 + \pi_{111} YoB_8 + \pi_{112} YoB_9   \\
&+ \pi_{113} Reg_1 + \pi_{114} Reg_2 + \pi_{115} Reg_3 + \pi_{116} Reg_4 + \pi_{117} Reg_5 \\
&+ \pi_{118} Reg_6 + \pi_{119} Reg_7 + \pi_{120} Reg_8  \\
&+ \pi_{121} Qob_1 + \pi_{122} Qob_2 + \pi_{123} Qob_3  &  \text{(3 variables)} \\
&+ \pi_{124} Qob_1 \times YoB_1 + \pi_{125} Qob_2 \times YoB_2 +\cdots + \pi_{140} Qob_3 \times YoB_9   & \text{(3*9=27 variables)} \\
&+ \eta_{1i} \\
\end{aligned}`
$$

- Second stage (structural equation):

$$
`\begin{aligned}
Ln(W_{i}) &= \delta_{0} +\delta_{1} \widehat{S}_{i} + \beta_{1} Race_i + \beta_{2} SMSA_i + \beta_{3} Married_i \\
&+ \beta_{4} YoB_1 + \beta_{5} YoB_2 + \beta_{6} YoB_3  + \beta_{7} YoB_4  + \beta_{8} YoB_5 \\
&+ \beta_{9} YoB_6 + \beta_{10} YoB_7 + \beta_{11} YoB_8 + \beta_{12} YoB_9 \\
&+ \beta_{13} Reg_1 + \beta_{14} Reg_2 + \beta_{15} Reg_3 + \beta_{16} Reg_4 + \beta_{17} Reg_5 \\
&+ \beta_{18} Reg_6 + \beta_{19} Reg_7 + \beta_{20} Reg_8 \\
&+ \epsilon_{i} \\
\end{aligned}`
$$

]

.footnote[
- Total numbers of excluded instruments is `$(27+3)=30$` for TSLS model in column (6)
]

---

### Model illustration: Column (7) OLS in Table V

The OLS model is:

$$
`\begin{aligned}
Ln(W_{i}) &= \delta_{0} +\delta_{1} S_{i} + \beta_{1} Race_i + \beta_{2} SMSA_i + \beta_{3} Married_i  \\
&+ \beta_{4} YoB_1 + \beta_{5} YoB_2 + \beta_{6} YoB_3  + \beta_{7} YoB_4  + \beta_{8} YoB_5 \\
&+ \beta_{9} YoB_6 + \beta_{10} YoB_7 + \beta_{11} YoB_8 + \beta_{12} YoB_9 \\
&+ \beta_{13} Reg_1 + \beta_{14} Reg_2 + \beta_{15} Reg_3 + \beta_{16} Reg_4 + \beta_{17} Reg_5 \\
&+ \beta_{18} Reg_6 + \beta_{19} Reg_7 + \beta_{20} Reg_8 \\
&+ \beta_{21} Age_i + \beta_{22} Age^2_i \\
&+ \epsilon_{i} \\
\end{aligned}`
$$

---

### Model illustration: Column (8) TSLS in Table V

.page-font-20[

- First stage (reduced equation):

- Second stage (structural equation):

$$
`\begin{aligned}
Ln(W_{i}) &= \delta_{0} +\delta_{1} \widehat{S}_{i} + \beta_{1} Race_i + \beta_{2} SMSA_i + \beta_{3} Married_i  \\
&+ \beta_{4} YoB_1 + \beta_{5} YoB_2 + \beta_{6} YoB_3  + \beta_{7} YoB_4  + \beta_{8} YoB_5 \\
&+ \beta_{9} YoB_6 + \beta_{10} YoB_7 + \beta_{11} YoB_8 + \beta_{12} YoB_9 \\
&+ \beta_{13} Reg_1 + \beta_{14} Reg_2 + \beta_{15} Reg_3 + \beta_{16} Reg_4 + \beta_{17} Reg_5 \\
&+ \beta_{18} Reg_6 + \beta_{19} Reg_7 + \beta_{20} Reg_8 \\
&+ \beta_{21} Age_i + \beta_{22} Age^2_i \\
&+ \epsilon_{i} \\
\end{aligned}`
$$

]

.footnote[
- Total numbers of excluded instruments is `$(27+3)-2=30-2=28$` for TSLS model in column (8)
]

---

### Model illustration: findings

All of the TSLS estimates we have presented so far are overidentified because several estimates of the return to education could be constructed from subsets of the instruments.

The `$\chi^{2}$` statistics presented at the bottom of Tables IV, V, and VI test the hypothesis that the various combinations of instruments yield the same estimate of the return to education.

- This statistic is calculated as the sample size times the `$R^{2}$` from a regression of the residuals from the TSLS equation on the exogenous variables and instruments [Newey, 1985].

- In spite of the huge sample sizes, the overidentifying restrictions (`$H_0$`) are not rejected in the models in Tables V.

---
exclude: true

### Second stage regression (results Table VI)

.page-font-20[

| Independent variable | (1) OLS | (2) TSLS | (3) OLS | (4) TSLS | (5) OLS | (6) TSLS | (7) OLS | (8) TSLS |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Years of education | 0.0573 (0.0003) | 0.0553 (0.0138) | 0.0573 (0.0003) | 0.0948 (0.0223) | 0.0520 (0.0003) | 0.0393 (0.0145) | 0.0521 (0.0003) | 0.0779 (0.0239) |
| Race ( 1 = black ) | -- | - | - | - | -0.2107 (0.0032) | -0.2266 (0.0183) | -0.2108 (0.0032) | -0.1786 (0.0296) |
| SMSA (1 = center city) | - | - | - | - | 0.1418 (0.0023) | 0.1535 (0.0135) | 0.1419 (0.0023) | 0.1182 (0.0220) |
| Married ( 1 = married) | - | - | - | - | 0.2445 (0.0022) | 0.2442 (0.0022) | 0.2444 (0.0022) | 0.2450 (0.0023) |
| 9 Year-of-birth dummies | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes |
| 8 Region-of-residence dummies | No | No | No | No | Yes | Yes | Yes | Yes |
| Age | - | - | 0.1800 (0.0389) | 0.1325 (0.0486) | - | - | 0.1518 (0.0379) | 0.1215 (0.0474) |
| Age-squared | - | - | 0.0023 (0.0006) | 0.0016 (0.0007) | - | - | 0.0019 (0.0005) | 0.0015 (0.0007) |
| `$\chi^{2} [dof]$` | - | 101.6 [29] | - | 49.1 [27] | - | 93.6 [29] | - | 50.6 [27] |

]

.footnote[
- **TABLE VI**: OLS and TSLS Estimates of the Return to Education for Men Born **1940-1949** : 1980 Census
- **Source**: Angrist and Krueger (1991)
]

---

### More instruments: add state-of-birth dummies

Although most schools admit students born in the beginning of the year at an older age, school start age policy varies across states and across school districts within many states.

Let's focus on age cohort 40-49(Men Born **1930-1939**), 1980 Census.

More instruments can increase variation in the predicted schooling variable, lowering standard errors and tightening confidence intervals.

So variability in education in 2SLS is solely due to differences in seasons of birth and this is allowed to vary by state and birth year for the first time.

---

### More instruments: ALL available instruments

AK1991 include more instruments in their two stage least squares model.

Available instruments:

- Nine year-of-birth dummies (9 variables)
- Three QoB dummies (3 variables)
- Fifty state-of-birth dummies (50 variables)
- Three QoB dummies interacted with 50 state-of-birth dummies (150 variables)
- Three QoB dummies interacted with 9 year-of-birth dummies (27 variables)

---

### More instruments: Excluded instruments (TABLE VII)

AK1991 specify totally 180 excluded instruments in stage 2 (model without the intercept term):

Notes for `TABLE VII`:

- Age and age-squared are measured in quarters of years.
- Each equation also includes an intercept term. 
- The sample size is 329,509.

Excluded instruments in TSLS models for `TABLE VII`:

- 188 for column (8)
- 186 for column (7)
- 180 for column (6)
- 178 for column (5)

---

### More instruments: OLS VS TSLS (age cohort 40-49)

.page-font-20[

| Independent variable | (1) OLS | (2) TSLS | (3) OLS | (4) TSLS | (5) OLS | (6) TSLS | (7) OLS | (8) TSLS |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Years of education | 0.0673 (0.0003) | 0.0928 (0.0093) | 0.0673 (0.0003) | 0.0907 (0.0107) | 0.0628 (0.0003) | 0.0831 (0.0095) | 0.0628 (0.0003) | 0.0811 (0.0109) |
| Race ( 1 = black ) | - | - | - | - | -0.2547 (0.0043) | -0.2333 (0.0109) | -0.2547 (0.0043) | -0.2354 (0.0122) |
| SMSA (1 = center city) | - | - | - | - | 0.1705 (0.0029) | 0.1511 (0.0095) | 0.1705 (0.0029) | 0.1531 (0.0107) |
| Married (1 = married) | - | - | - | - | 0.2487 (0.0032) | 0.2435 (0.0040) | 0.2487 (0.0032) | 0.2441 (0.0042) |
| 9 Year-of-birth dummies | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes |
| 8 Region-of-residence dummies | No | No | No | No | Yes | Yes | Yes | Yes |
| 50 State-of-birth dummies | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes |
| Age | - | - | -0.0757 (0.0617) | -0.0880 (0.0624) | - | - | -0.0778 (0.0603) | -0.0876 (0.0609) |
| Age-squared | - | - | 0.0008 (0.0007) | 0.0009 (0.0007) | - | - | 0.0008 (0.0007) | 0.0009 (0.0007) |
| `$\chi^{2}$` [dof] | - | 163 [179] | - | 161 [177] | - | 164 [179] | - | 162 [177] |
| .red[Excluded instruments] | | 188 | | 186 | | 180 | | 178 |

]

.footnote[
- **TABLE VII**: OLS and TSLS Estimates of the Return to Education for Men Born **1930-1939** : 1980 Census
- **Source**: Angrist and Krueger (1991)
]

---

### Model illustration: Column (7) OLS in Table VII

The OLS model is:

$$
`\begin{aligned}
Ln(W_{i}) &= \delta_{0} +\delta_{1} S_{i} + \beta_{1} Race_i + \beta_{2} SMSA_i + \beta_{3} Married_i  \\
&+ \beta_{4} YoB_1 + \beta_{5} YoB_2 + \beta_{6} YoB_3  + \beta_{7} YoB_4  + \beta_{8} YoB_5 \\
&+ \beta_{9} YoB_6 + \beta_{10} YoB_7 + \beta_{11} YoB_8 + \beta_{12} YoB_9 \\
&+ \beta_{13} Reg_1 + \beta_{14} Reg_2 + \beta_{15} Reg_3 + \beta_{16} Reg_4 + \beta_{17} Reg_5 \\
&+ \beta_{18} Reg_6 + \beta_{19} Reg_7 + \beta_{20} Reg_8 \\
&+ \beta_{21} Age_i + \beta_{22} Age^2_i \\
&+ \beta_{23} State_{1} + \beta_{24} State_{2} + \cdots + \beta_{72} State_{50}  & \text{(50 variables)} \\
&+ \epsilon_{i} \\
\end{aligned}`
$$

---

### Model illustration: Column (8) TSLS in Table VII

.page-font-18[

- First stage (reduced equation):

- Second stage (structural equation):

$$
`\begin{aligned}
Ln(W_{i}) &= \delta_{0} +\delta_{1} \widehat{S}_{i} + \beta_{1} Race_i + \beta_{2} SMSA_i + \beta_{3} Married_i  \\
&+ \beta_{4} YoB_1 + \beta_{5} YoB_2 + \beta_{6} YoB_3  + \beta_{7} YoB_4  + \beta_{8} YoB_5 \\
&+ \beta_{9} YoB_6 + \beta_{10} YoB_7 + \beta_{11} YoB_8 + \beta_{12} YoB_9 \\
&+ \beta_{13} Reg_1 + \beta_{14} Reg_2 + \beta_{15} Reg_3 + \beta_{16} Reg_4 + \beta_{17} Reg_5 \\
&+ \beta_{18} Reg_6 + \beta_{19} Reg_7 + \beta_{20} Reg_8 \\
&+ \beta_{21} Age_i + \beta_{22} Age^2_i \\
&+ \beta_{23} State_{1} + \beta_{24} State_{2} + \cdots + \beta_{72} State_{50}  & \text{(50 variables)} \\
&+ \epsilon_{i} \\
\end{aligned}`
$$

]

.footnote[
- Total numbers of excluded instruments is `$(27+3)+150-2=30+150-2=178$` for TSLS model in column (8)
]

---

### More instruments: alternative dependent variables

In addition to the log weekly wage, AK1991 have also examined the impact of compulsory schooling `$S_i$` on the log of annual salary and on weeks worked `$Ln(Wks)_i$`.

This exercise suggests that the main impact of compulsory schooling is on the log weekly wage `$Ln(W)_i$`, and not on weeks worked `$Ln(Wks)_i$`.

For example, when the log of weeks worked `$Ln(Wks)_i$` is used as the dependent variable in column (5) and column (6) of Table VII instead of the log weekly wage `$Ln(W)_i$`

- The TSLS estimate of the return to education (**column (6)**) is 0.016 (less than 0.0831 in Table VII) with a standard error of 0.008 .

- This is within sampling variance of the OLS estimate of the return to education (**column (5)**), which is 0.008 (much less than 0.0628 in Table VII) with a standard error of 0.0002 .

---
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## Weak Instruments Problem

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---

### Weak Instruments Problem

- Important paper suggesting OLS and 2SLS were pretty similar, plus introduces modern notion of seeking "plausibly exogenous instruments"
- But in the early 1990s, a number of papers showed that IV can be severely biased with weak instruments and many instruments for one endogenous variable
- In the worst case, if the instruments are so weak that there is no first stage, then the 2SLS sampling distribution is centered on the probability limit of OLS

---

### Matrices and instruments

- The causal model of interest is:

$$
Y=\beta X+\nu
$$

- Matrix of instrumental variables is `$Z$` with the first stage equation:

$$
X=Z^{\prime} \pi+\eta
$$

---

### Weak instruments and bias towards OLS

- If 
`$\nu_{i}$` from causal model and 
`$\eta_{i}$` from first stage model are correlated, then OLS estimated 
`$\widehat{\beta}_{O L S}$` in causal model is biased
- To show the bias of OLS, take the population mean difference in 
`$\beta$` minus estimated 
`$\beta_{O L S}$` :

$$
`\begin{aligned}
E\left[\widehat{\beta}_{O L S}-\beta\right] 
 =\frac{\operatorname{Cov}(\nu, X)}{\operatorname{Var}(X)}
 =\frac{\sigma_{\nu \eta}}{\sigma_{\eta}^{2}}
\end{aligned}`
$$

- Our hope is that with 2SLS, we can drive this bias to zero in the finite sample and have a reasonably unbiased estimate of `$\beta$`

---

### Weak instruments and 2SLS bias towards OLS

- Strong instruments shrink the bias term, 
`$\frac{\sigma_{\nu \eta}}{\sigma_{\eta}^{2}}$`, to an inconsequential scaled value (but cannot go to zero)
- We can derive the approximate bias of 2SLS as:

$$
`\begin{aligned}
E\left[\widehat{\beta}_{2 S L S}-\beta\right] 
& \approx \frac{\sigma_{\nu \eta}}{\sigma_{\eta}^{2}} \frac{1}{F+1}
\end{aligned}`
$$

- Consider the intuition all that work bought us now: if the first stage is weak (i.e, 
`$F \rightarrow 0$` ), then the bias of 2SLS approaches 
`$\frac{\sigma_{\nu \eta}}{\sigma_{\eta}^{2}}$`

---

### Weak instruments and bias towards OLS

- This is the same as the OLS bias as for 
`$\pi=0$` in the second equation on the earlier slide (i.e., there is no first stage relationship) `$\sigma_{x}^{2}=\sigma_{\eta}^{2}$` and therefore the OLS bias `$\frac{\sigma_{\nu \eta}}{\sigma_{\eta}^{\eta}}$` becomes `$\frac{\sigma_{\nu \eta}}{\sigma_{\eta}^{2}}$`.

- But if the first stage is very strong 
`$(F \rightarrow \infty)$` then the 2SLS bias is approaching 0 .

- Cool thing is - you can test this with an F test on the joint significance of 
`$Z$` in the first stage

- It's absolutely critical therefore that you choose instruments that are strongly correlated with the endogenous regressor, otherwise the cure is worse than the disease

---

### Weak Instruments - Adding More Instruments

- Adding more weak instruments will increase the bias of 2SLS
`$\rightarrow$` By adding further instruments without predictive power, the first stage `$F$`-statistic goes toward zero and the bias increases
`$\rightarrow$` We will see this more closely when we cover the leniency design
- If the model is "just identified" - mean the same number of instrumental variables as there are endogenous covariates - weak instrument bias is less of a problem

---

### Weak instrument problem

- After Angrist and Krueger study, there were new papers highlighting issues related to weak instruments and finite sample bias
- Key papers are Nelson and Startz (1990), Buse (1992), Bekker (1994) and especially Bound, Jaeger and Baker (1995)
- Bound, Jaeger and Baker (1995) (BJB1995) highlighted this problem for the Angrist and Krueger study.

---

### Bound, Jaeger and Baker (1995)

Remember, AK1991 present findings from expanding their instruments to include many interactions (i.e., saturated model)

1. Quarter of birth dummies `$\rightarrow 3$` instruments
2. Quarter of birth dummies + (quarter of birth) `$\times$` (year of birth) + (quarter of birth) 
`$\rightarrow$` 183 instruments
3. Thus totally  `$183-3=180$` excluded instruments.

So if any of these are weak, then the approximate bias of 2SLS gets worse.

---

### Instruments in Angrist and Krueger (model setup)

Calculated from the 5% Public-Use Sample of the 1980 U.S. Census for men **born 1930-1939**

- Sample size is 329,509.

- All specifications include Race (1 = black), SMSA (1 = central
 city), Maried (1 = maried, living with spouse), 8 Regional dummies.
- F (first stage) and partial 
`$R^2$` are for the instruments in the first stage of IV estimation. 
- F (overidentification) is that suggested by Basmann (1960).

---

### Instruments in Angrist and Krueger (results)

.page-font-20[

| | (1) OLS | (2) IV | (3) OLS | (4) IV | (5) OLS | (6) IV |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: |
| | | | | | |
| Coefficient | .063 (.000) | .142 (.033) | .063 (.000) | .081 (.016) | .063 (.000) | .060 (.029) |
| `$F$` (excluded instruments) | | 13.486 | | 4.747 | | 1.613 |
| Partial `$R^{2}$` (excluded instruments, `$\times 100$` ) | | . 012 | | . 043 | | . 014 |
| `$F$` (overidentification) | | . 932 | | . 775 | | . 725 |
| **Age Control Variables**: | | | | |
| Age, Age `${ }^{2}$` | `$\checkmark$` | `$\checkmark$` | | | `$\checkmark$` | `$\checkmark$` |
| 9 Year of birth dummies | | | `$\checkmark$` | `$\checkmark$` | `$\checkmark$` | `$\checkmark$` |
| **Excluded Instruments**: | | | | | |
| Quarter of birth | | `$\checkmark$` | | `$\checkmark$` | | `$\checkmark$` |
| Quarter of birth `$\times$` state of birth | | | | `$\checkmark$` | | `$\checkmark$` |
| Number of excluded instruments | | 3 | | 30 | | 28 |

]

.footnote[
- Source: Bound, Jaeger and Baker (1995)
- Table 1. Estimated Effect of Completed Years of Education on Men's Log Weekly Earnings
- Standard errors of coefficients in parentheses
]

---

### Instruments in Angrist and Krueger (findings)

Adding more weak instruments reduced the first stage `$F$`-statistic and increases the bias of 2SLS. Notice its also moved closer to OLS.

- The 
`$F$` statistic (excluded instruments)  decreases across the different IV specifications on .red[column (2),（4）] and .red[column (6)], suggesting negligible finite-sample bias.

- The IV specification using within-year age controls .red[column (6)] (coef.=0.060) to be more sensible than the specification that does not .red[column (4)] (coef.=0.081).

- Comparing the partial `$R^{2}$` in .red[columns (2)] and .red[column (6)] shows that adding 25 instruments does not change the explanatory power of the excluded instruments by very much.

> Explained that why the `$F$` statistic deteriorates so much between the two specifications.

---

### Adding more instruments: state of birth

Calculated from the 5% Public-Use Sample of the 1980 U.S. Census for men **born 1930-1939**

- Sample size is 329,509.

- All specifications include Race (1 = black), SMSA (1 = central
 city), Maried (1 = maried, living with spouse), 8 Regional dummies, and .red[50 State of Birth dummies]
 as control variables.
- F (first stage) and partial 
`$R^2$` are for the instruments in the first stage of IV estimation. 
- F (overidentification) is that suggested by Basmann (1960).

- We have .red[more instruments] now, including 50 `state of birth` and interaction terms.

---

### Adding more instruments: state of birth (results)

.page-font-20[

| | (1) OLS | (2)IV | (3) OLS | (4) IV |
| :--- | :---: | :---: | :---: | :---: |
| Coefficient | .063 (.000) | .083 (.009) | .063 (.000) | .081 (.011) |
| `$F$` (excluded instruments) | | 2.428 | | 1.869 |
| Partial `$R^{2}$` (excluded instruments, `$\left.\times 100\right)$` | | .133 | | .101 |
| `$F$` (overidentification) | | .919 | | .917 |
| **Age Control Variables**|
| Age, `$\mathrm{Age}^{2}$`| | | `$\checkmark$` | `$\checkmark$` |
| 9 Year of birth dummies| `$\checkmark$` | `$\checkmark$` | `$\checkmark$` | `$\checkmark$` |
| **Excluded Instruments**|
| Quarter of birth | | `$\checkmark$` | | `$\checkmark$` |
| Quarter of birth `$\times$` year of birth | | `$\checkmark$` | | `$\checkmark$` |
| Quarter of birth `$\times$` state of birth | | `$\checkmark$` | | `$\checkmark$` |
| Number of excluded instruments| | 180 | | 178 |

]

.footnote[
- Source: Bound, Jaeger and Baker (1995)
- Table 2. Estimated Effect of Completed Years of Education on Men's Log Weekly Earnings, Controlling for State of Birth
- Standard errors of coefficients in parentheses
]

---

### Adding more instruments: state of birth (findings)

More instruments increase precision, but drive down `$F$`, therefore we know the problem has gotten worse.

- F-statistic (excluded instruments) is 1.869 in .red[column (4)], which is much lower than 2.428 in .red[column (2)].

---

### Randomly generated instruments (model setup)

To illustrate that second-stage results do not give us any indication of the existence of quantitatively important finitesample biases, we reestimated Table 1, columns (4) and (6), and Table 2 , columns (2) and (4), using randomly generated information in place of the actual quarter of birth, following a suggestion by Alan Krueger.

- BJB1995 use .red[randomly generated information] in place of the actual quarter of birth.

---

### Randomly generated instruments (results)

| Table (column)             | T1 (c4)  | T1 (c6)  | T2 (c2) | T2 (c4) |
|:---------------------------|:--------:|:--------:|:-------:|:-------:|
| **Estimated Coefficient**:      |          |          |         |         |
| Mean                       | . 062    | . 061    | . 060   | . 060   |
| Standard deviation of mean | . 038  | . 039  | . 015 | . 015 |
| 5th percentile             | -. 001 | -. 002 | . 034 | . 035 |
| Median                     | . 061  | . 061  | . 060 | . 060 |
| 95th percentile            | . 119  | . 127  | . 083 | . 082 |
| **Estimated Standard Error**:   |        |        |       |       |
| Mean                       | . 037  | . 039  | . 015 | . 015 |

.footnote[
- Source: Bound, Jaeger and Baker (1995)
- Table 3. Estimated Effect of Completed Years of Education on Men's Log Weekly Earnings, Using Simulated Quarter of Birth (500 replications)
]

---

### Adding more instruments: state of birth (findings)

It is striking that the second-stage results reported in Table 3 look quite reasonable even with no information about educational attainment (here, quarter of birth) in the simulated instruments.

They give no indication that the instruments were randomly generated.

- The mean of the .red[estimated coefficients] in each case is close to the comparable OLS estimate (compare to Table 1 and Table 2).

- The mean of the .red[estimated standard errors] of the coefficients (Table 3) are quite close to the actual standard deviations of the 500 estimates for each model.

BJB (Bound, Jaeger and Baker, 1995) conclude that it is likely that some of the results reported in AK (Angrist and Krueger, 1991) are affected by quantitatively large finitesample biases.

---

### IV advice: learn more about Weak instruments

- Excellent review by Keane and Neal (2024) "A Practical Guide to Weak Instruments" as well as Andrews, Stock and Sun (2019)

- Stock, Wright and Yogo (2002) found that `$F$` statistics on the excludability of the instrument from the first stage .red[greater than 10] performed well in Monte Carlos with homoskedasticity, but 2SLS is has poor properties here

-  Under powered
    -  Artificially low standard errors when endogeneity is severe
    -  This causes `$t$`-tests to be misleading

.footnote[
- Keane, M. P., and T. Neal. A Practical Guide to Weak Instruments[J]. Annual Review of Economics, 2024, 16:185-212.
- Andrews, I., J. H. Stock, and L. Sun. Weak Instruments in Instrumental Variables Regression: Theory and Practice[J]. Annual Review of Economics, 2019, 11:727-753.

]

---

### IV advice: Weak instruments

> Andrews, Stock and Sun (2019):

> - In the leading case with a single endogenous regressor, we recommend that researchers judge instrument strength based on the effective F-statistic of Montiel Olea and Pflueger (2013).

> - If there is a single instrument, we recommend reporting identification robust Anderson-Rubin confidence intervals. These are effective regardless of the strength of the instruments, and so should be reported regardless of the value of the first stage F.

> - Finally, if there are multiple instruments, the literature has not yet converged on a single procedure, but we recommend choosing from among the several available robust procedures that are efficient when the instruments are strong.

---

### IV advice: Weak instruments

> - Anderson-Rubin greatly alleviate this problem and should be used even with very strong instruments provided the first-stage `$F$` is well above 10 (Lee, et al. 2020 say 104.7)

> - Higher thresholds are recommended, and even then .red[robust tests] are suggested unless `$F$` is in the thousands

> - Keane and Neal (2024) write, "to avoid over-rejecting the null when `$\beta_{2SLS}$` is shifted in the direction of the OLS bias, one should rely on the Anderson-Rubin test rather than the `$t$`-test even when the first-stage `$F$`-statistic is in the thousands."

---
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class: inverse,center

# End Of This Session