1. Hypothesis Testing & Inference (Week 7)
Key Assumptions (MLR.1–MLR.6)
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MLR.1–MLR.4: OLS is unbiased.
MLR.5: Homoskedasticity:
Var(𝑢 ∣ 𝑥! , … , 𝑥" ) = 𝜎 #
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MLR.6: Normality of errors:
𝑢 ∼ 𝑁(0, 𝜎 # )(i.i.d.)
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Under MLR.1–MLR.6, OLS estimators are normally distributed:
𝛽/$ ∼ 𝑁0𝛽$ ,Var(𝛽/$ )1
Variance of OLS Estimator
Var(𝛽/$ ) =
𝜎#
𝑇𝑆𝑆$ (1 − 𝑅$# )
where 𝑇𝑆𝑆$ = ∑(𝑥$ − 𝑥¯$ )# , 𝑅$# = R² from regressing 𝑥$ on other 𝑥's.
Estimating 𝜎 !
&
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𝜎9 =
< 𝑢9%#
𝑛−𝑘−1
#
%'!
Standard error:
se(𝛽/$ ) =
𝜎9
=𝑇𝑆𝑆$ (1 − 𝑅$# )
t-statistic
𝑡=
𝛽/$ − 𝛽$
∼ 𝑡&("(!
se(𝛽/$ )
Testing Single Parameter
1. Null hypothesis: 𝐻) : 𝛽$ = 𝑎 (usually 𝑎 = 0).
2. Alternative:
Two-sided: 𝐻! : 𝛽$ ≠ 𝑎
o One-sided: 𝐻! : 𝛽$ > 𝑎 or 𝐻! : 𝛽$ < 𝑎
3. Compute t-statistic:
o
𝑡=
𝛽/$ − 𝑎
se(𝛽/$ )
4. Compare to critical value 𝑐 from 𝑡&("(! :
Two-sided: Reject if ∣ 𝑡 ∣> 𝑐
o One-sided: Reject if 𝑡 > 𝑐 (right) or 𝑡 < −𝑐 (left)
5. p-value: Probability of observing 𝑡 or more extreme under 𝐻) .
o
Confidence Interval for 𝛽"
𝛽/$ ± 𝑐 ⋅ se(𝛽/$ )
where 𝑐 = 𝑡*/#,&("(! .
Testing Linear Combination of Parameters
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Example: 𝐻) : 𝛽! = 𝛽# → test 𝐻) : 𝛽! − 𝛽# = 0.
t-statistic:
𝑡=
𝛽/! − 𝛽/#
se(𝛽/! − 𝛽/# )
where se(𝛽/! − 𝛽/# ) = =Var(𝛽/! ) + Var(𝛽/# ) − 2Cov(𝛽/! , 𝛽/# ).
Alternative: Reparameterize: define 𝜃 = 𝛽! − 𝛽# , rewrite model, test 𝐻) : 𝜃 = 0.
F-test for Multiple Restrictions
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Null: 𝐻) : 𝛽"(-.! = ⋯ = 𝛽" = 0 (q restrictions).
Unrestricted model: includes all variables.
Restricted model: excludes the q variables.
F-statistic:
𝐹=
(𝑅𝑆𝑆/ − 𝑅𝑆𝑆0/ )/𝑞
∼ 𝐹-,&("(!
𝑅𝑆𝑆0/ /(𝑛 − 𝑘 − 1)
Equivalently:
𝐹=
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#
(𝑅0/
− 𝑅/# )/𝑞
# )/(𝑛 − 𝑘 − 1)
(1 − 𝑅0/
Reject if 𝐹 > 𝐹*,-,&("(! .
Chow Test for Structural Break
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Test if regression function differs across groups.
Unrestricted: Separate regressions for each group.
Restricted: Pooled regression.
F-statistic:
𝐹=
[𝑅𝑆𝑆1 − (𝑅𝑆𝑆! + 𝑅𝑆𝑆# )]/(𝑘 + 1)
(𝑅𝑆𝑆! + 𝑅𝑆𝑆# )/[𝑛 − 2(𝑘 + 1)]
2. Functional Forms (Week 8)
Data Scaling & Standardization
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Rescaling y by c: coefficients multiplied by c.
Rescaling x by c: coefficient divided by c.
Standardized coefficients (beta coefficients):
𝑏R$ =
𝜎9$
𝛽/
𝜎92 $
Interpretation: change in y (in std dev) for 1 std dev change in 𝑥$ .
Quadratic Terms
Model:
𝑦 = 𝛽) + 𝛽! 𝑥 + 𝛽# 𝑥 # + 𝑢
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Marginal effect:
Δ𝑦
≈ 𝛽! + 2𝛽# 𝑥
Δ𝑥
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Turning point:
𝑥∗ = −
𝛽!
2𝛽#
Interaction Terms
Model:
𝑦 = 𝛽) + 𝛽! 𝑥! + 𝛽# 𝑥# + 𝛽4 (𝑥! × 𝑥# ) + 𝑢
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Effect of 𝑥! on y:
Δ𝑦
= 𝛽! + 𝛽4 𝑥#
Δ𝑥!
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Effect of 𝑥# on y:
Δ𝑦
= 𝛽# + 𝛽4 𝑥!
Δ𝑥#
Partial Effect at the Mean
Center variables at mean:
𝑦 = 𝛼) + 𝛿! 𝑥! + 𝛿# 𝑥# + 𝛽4 (𝑥! − 𝑥¯! )(𝑥# − 𝑥¯# ) + 𝑢
Then 𝛿# = effect of 𝑥# when 𝑥! = 𝑥¯! .
3. Categorical Variables (Week 9)
Binary (Dummy) Variables
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Base group: omitted category.
Intercept shift: coefficient on dummy = difference in intercept relative to base.
Avoid dummy variable trap: For g categories, include g−1 dummies.
Interaction with Dummies
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Different intercepts:
𝑦 = 𝛽) + 𝛿) 𝐷 + 𝛽! 𝑥 + 𝑢
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Different slopes:
𝑦 = 𝛽) + 𝛿) 𝐷 + 𝛽! 𝑥 + 𝛿! (𝐷 × 𝑥) + 𝑢
Slope for 𝐷 = 1: 𝛽! + 𝛿! .
Ordinal Variables
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Option 1: Include as continuous (assumes linear effect).
Option 2: Create dummies for each category (more flexible).
Log-Dummy Models
Model:
ln (𝑦) = 𝛽) + 𝛿𝐷 + 𝛽! 𝑥 + 𝑢
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Approximate % change: 100 ⋅ 𝛿%
Exact % change: 100 ⋅ [exp (𝛿) − 1]%
4. Binary Dependent Variable (Week 9)
Linear Probability Model (LPM)
𝑃(𝑦 = 1 ∣ 𝑥) = 𝛽) + 𝛽! 𝑥! + ⋯ + 𝛽" 𝑥"
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Interpretation: 𝛽$ = change in probability that y=1 for 1-unit increase in 𝑥$ .
Issues:
1. Predicted probabilities may be outside [0,1].
2. Heteroskedasticity:
Var(𝑢 ∣ 𝑥) = 𝑝(𝑥)[1 − 𝑝(𝑥)]
→ Use robust standard errors.
5. Heteroskedasticity (Week 10)
Consequences of Heteroskedasticity
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OLS still unbiased, consistent.
Usual SEs biased → invalid t/F tests.
OLS not BLUE.
Robust Standard Errors (White/Huber)
For simple regression:
b (𝛽/! ) =
Var
∑(𝑥% − 𝑥¯)# 𝑢9%#
𝑇𝑆𝑆5#
In general:
b (𝛽/$ ) =
Var
where 𝑟̂%$ = residual from regressing 𝑥$ on other x's.
∑𝑟̂%$# 𝑢9%#
𝑅𝑆𝑆$#
Testing for Heteroskedasticity
Breusch-Pagan Test
1. Regress 𝑦 on 𝑥's, get residuals 𝑢9.
2. Regress 𝑢9# on 𝑥's, get 𝑅0#6! .
3. LM statistic:
𝐿𝑀 = 𝑛 ⋅ 𝑅0#6! ∼ 𝜒"#
4. F-statistic:
𝐹=
𝑅0#6! /𝑘
01 − 𝑅06#! )/(𝑛 − 𝑘 − 11
∼ 𝐹",&("(!
White Test
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Auxiliary regression: 𝑢9# on 𝑥's, squares, and cross-products.
Or modified: 𝑢9# on 𝑦9 and 𝑦9 # (2 restrictions).
Weighted Least Squares (WLS)
If Var(𝑢 ∣ 𝑥) = 𝜎 # ℎ(𝑥), transform:
𝑦
iℎ(𝑥)
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= 𝛽)
1
iℎ(𝑥)
+ 𝛽!
𝑥!
iℎ(𝑥)
+ ⋯+
𝑢
iℎ(𝑥)
GLS estimator = OLS on transformed equation.
More efficient than OLS if variance form correctly specified.
Prediction with Log Models
If model is: ln (𝑦) = 𝛽) + 𝛽! 𝑥! + ⋯ + 𝛽" 𝑥" + 𝑢
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b𝑦) is biased.
Naïve prediction: 𝑦9 = exp (ln
Bias-corrected (assuming normality):
𝜎9 #
b𝑦)
𝑦9 = exp j k ⋅ exp (ln
2
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Duan’s smearing estimator (robust):
𝑎9) =
1
b𝑦)
∑exp (𝑢9% ), 𝑦9 = 𝑎9) ⋅ exp (ln
𝑛
6. Program Evaluation (Week 9)
Potential Outcomes Framework
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𝑦% (1): outcome if treated.
𝑦% (0): outcome if not treated.
Treatment effect: 𝑇𝐸% = 𝑦% (1) − 𝑦% (0).
Observed outcome: 𝑦% = 𝑥% 𝑦% (1) + (1 − 𝑥% )𝑦% (0).
Average Treatment Effect (ATE)
𝜏789 = 𝐸[𝑦(1) − 𝑦(0)]
If constant treatment effect:
where 𝜏 = 𝜏789 .
𝑦 = 𝛼) + 𝜏𝑥 + 𝑢
Assumptions for Causal Inference
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Random assignment: 𝑥 ⊥ (𝑦(0), 𝑦(1)).
Unconfoundedness: 𝑥 ⊥ (𝑦(0), 𝑦(1)) ∣ 𝑧 (conditional on covariates).
Quick Reference Formulas
OLS variance
Var(𝛽/$ ) = 𝜎 # /[𝑇𝑆𝑆$ (1 − 𝑅$# )]
t-statistic
𝑡 = (𝛽/$ − 𝑎)/se(𝛽/$ )
Confidence interval
𝛽/$ ± 𝑡*/# ⋅ se(𝛽/$ )
F-statistic
𝐹=
(𝑅𝑆𝑆/ − 𝑅𝑆𝑆0/ )/𝑞
𝑅𝑆𝑆0/ /(𝑛 − 𝑘 − 1)
BP test LM
𝐿𝑀 = 𝑛 ⋅ 𝑅0#6! ∼ 𝜒"#
WLS transformation
divide by iℎ(𝑥)
Log-dummy % change
exact: 100[exp (𝛿) − 1]%
LPM interpretation
𝛽$ = Δ𝑃(𝑦 = 1 ∣ 𝑥)
Step-by-Step Procedures
1. Conduct a t-test
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State 𝐻) : 𝛽$ = 𝑎 and 𝐻! .
Compute 𝑡 = (𝛽/$ − 𝑎)/se(𝛽/$ ).
Find critical value 𝑐 from 𝑡&("(! at significance level 𝛼.
Compare: reject if ∣ 𝑡 ∣> 𝑐 (two-sided) or 𝑡 > 𝑐 (right) or 𝑡 < −𝑐 (left).
Report p-value.
2. Conduct an F-test
1. Estimate unrestricted and restricted models.
2. Compute 𝑅𝑆𝑆0/ and 𝑅𝑆𝑆/ .
(;<< (;<<#" )/3. Compute 𝐹 = ;<< " /(&("(!)
.
#"
4. Compare to 𝐹*,-,&("(! .
5. Reject if 𝐹 > 𝐹critical .
3. Test for Heteroskedasticity (BP)
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Run OLS, get residuals 𝑢9.
Regress 𝑢9# on all x's.
Compute 𝐿𝑀 = 𝑛 ⋅ 𝑅0#6! .
Compare to 𝜒"# critical value.
4. Implement WLS
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Assume/formulate variance function ℎ(𝑥).
Transform all variables: divide by iℎ(𝑥).
Run OLS on transformed equation.
Interpret coefficients in original units.
5. Predict with Log Model
b𝑦 and 𝜎9 # .
1. Estimate ln (𝑦) model, get ln
!
2. Compute 𝑎9) = ∑exp (𝑢9% ).
&
b𝑦).
3. Predicted 𝑦 = 𝑎9) ⋅ exp (ln
Common Pitfalls & Reminders
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Dummy variable trap: Omit one category.
Interaction terms: Always include main effects.
Log models: % interpretation; use exact formula for large coefficients.
LPM: Use robust SEs; predictions outside [0,1] possible.
Heteroskedasticity: Always check; use robust SEs as default.
Chow test: Requires same error variance across groups.
ATE estimation: Random assignment is key for causality.