Large-Sample Properties of OLS

Topic 7 · Consistency and asymptotic normality

Prof. Dr. Hüseyin Taştan

Yıldız Technical University · Department of Economics

Fall 2026

Today’s roadmap

By the end of this lecture you should be able to

• distinguish finite-sample from asymptotic properties
• define consistency and explain why OLS is consistent under MLR.4′
• compute the asymptotic bias caused by an omitted variable
• state asymptotic normality and explain why t and F tests survive without normal errors
• compute the LM statistic and compare it with the F test
• explain asymptotic efficiency

Reading

Fall 2026 · Week 6

Wooldridge ch. 5 Heiss ch. 5

Lab

Simulations in R. See Lab 08 on the course page.

Previously

Unbiasedness, BLUE and exact t/F distributions all hold for any sample size n, but the last one needs normal errors.

Warm-up: what does a big sample fix?

A researcher has 50,000 observations. The errors are clearly non-normal, and she suspects an important variable is missing from the model.

Which of the two problems does the large sample take care of?

A) Both: with 50,000 observations everything works

B) Only the non-normality: t and F tests become approximately valid, but the omitted variable still biases the estimates

C) Neither: without normal errors nothing can be done

This is the whole topic in one sentence. A large sample buys us consistency and approximate normality of the estimators, so inference works without MLR.6. It does nothing about a violation of MLR.4: an omitted variable makes OLS inconsistent, and more data only makes us more confident about the wrong number.

Finite-sample vs. asymptotic properties

Finite-sample (any n)

• Unbiasedness (MLR.1–4)
• BLUE (MLR.1–5)
• Exact t and F distributions (MLR.1–6, including normality)

Asymptotic (n \to \infty)

• Consistency
• Asymptotic normality: approximate t and F distributions without MLR.6
• Asymptotic efficiency

What happens to the sampling distribution of \hat\beta_j as we collect more and more data?

Consistency

What happens to \hat\beta_j as the sample grows, and when more data does not help.

Consistency

Definition

An estimator W_n of \theta is consistent if, for every \varepsilon > 0, \;P(|W_n - \theta| > \varepsilon) \to 0 as n \to \infty. We write \text{plim}(W_n) = \theta.

• Example: the sample mean \bar Y is consistent for \mu (law of large numbers).
• Each n gives a sampling distribution. Consistency means these distributions pile up at the true value as n grows.

Theorem 5.1: consistency of OLS

Under MLR.1–MLR.4, \hat\beta_j is consistent for \beta_j. MLR.4 can be replaced by the weaker

MLR.4′: \;E(u) = 0 and \text{Cov}(x_j, u) = 0 for j = 1, \dots, k.

MLR.4′ is enough for consistency but not for unbiasedness.

Interactive: consistency

Why OLS is consistent, and when it is not

Simple regression: substitute y_i = \beta_0 + \beta_1 x_i + u_i and divide numerator and denominator by n:

\hat\beta_1 = \beta_1 + \frac{n^{-1}\sum_i (x_i - \bar x)\,u_i}{n^{-1}\sum_i (x_i - \bar x)^2} \;\xrightarrow{\;p\;}\; \beta_1 + \frac{\text{Cov}(x, u)}{\text{Var}(x)}

• Under MLR.4′, \text{Cov}(x, u) = 0, so \text{plim}\,\hat\beta_1 = \beta_1.
• Otherwise the asymptotic bias (inconsistency) is \;\text{Cov}(x_1, u)/\text{Var}(x_1).

Omitted variable: if y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + v and we omit x_2:

\text{plim}\,\tilde\beta_1 = \beta_1 + \beta_2\,\delta_1, \qquad \delta_1 = \frac{\text{Cov}(x_1, x_2)}{\text{Var}(x_1)}

Increasing the sample size does not cure inconsistency. \tilde\beta_1 just gets closer and closer to the wrong value.

Check: a trash incinerator

y: house price, x_1: distance to a trash incinerator, x_2: house quality (omitted). Assume \beta_1 > 0 and \beta_2 > 0.

If better houses tend to be built farther from the incinerator, the simple regression estimate of \beta_1 converges to…

A) a value larger than \beta_1: the effect of distance is overstated

B) a value smaller than \beta_1

C) \beta_1, because the sample is large

\delta_1 > 0 (quality rises with distance) and \beta_2 > 0, so \text{plim}\,\tilde\beta_1 = \beta_1 + \beta_2\delta_1 > \beta_1. A large sample does not help.

Large-sample inference

Why t, F and LM tests are approximately valid even when the errors are not normal.

Asymptotic normality

Theorem 5.2

Under the Gauss–Markov assumptions MLR.1–MLR.5, without normality:

\frac{\hat\beta_j - \beta_j}{\text{se}(\hat\beta_j)} \;\stackrel{a}{\sim}\; \text{Normal}(0, 1)

• The usual t and F tests and confidence intervals are approximately valid in large samples even if u is not normal. We only need a finite error variance.
• Homoskedasticity (MLR.5) and zero conditional mean (MLR.4) are still needed.
• Standard errors shrink at the rate 1/\sqrt{n}: four times the data halves the standard errors.

Interactive: do we need normal errors?

Start with “very skewed” at n = 10: the histogram of t statistics is lopsided. The left-tail test rejects the true H0 too rarely (about 3%) and the right-tail test too often (about 9%). The two-sided test is closer to 5% because the two errors partly cancel. Increase n: by about 1,000 observations all three rejection rates are close to 5%. With normal errors the size is right at every n. Monte Carlo noise is about ±0.5 percentage points.

Large samples in practice: critical values

The t_{n-k-1} distribution is the normal distribution once n is large. Two-sided 5% critical values:

round(c(df_10 = qt(.975, 10), df_30 = qt(.975, 30), df_120 = qt(.975, 120),
        df_1000 = qt(.975, 1000), normal = qnorm(.975)), 3)

#>   df_10   df_30  df_120 df_1000  normal 
#>   2.228   2.042   1.980   1.962   1.960

By about 120 degrees of freedom the difference is in the third decimal. This is why, with a big sample, we can quote 1.96 and stop worrying about degrees of freedom — and why the F and LM tests agree.

The Lagrange multiplier (LM) test

A large-sample alternative to the F test for q exclusion restrictions:

\begin{aligned} H_0&: \beta_{k-q+1} = 0,\; \dots,\; \beta_k = 0\\ H_1&: \beta_j \neq 0 \;\text{ for at least one } j = k-q+1, \dots, k \end{aligned}

1. Estimate the restricted model and save the residuals \tilde u.
2. Regress \tilde u on all the regressors, including the excluded ones. Save its R^2_{\tilde u}.
3. Compute \;LM = n\,R^2_{\tilde u}.

Under H_0 and MLR.1–MLR.5 (normality is not needed), \;LM \stackrel{a}{\sim} \chi^2_q. Reject H_0 if LM > c, with c from \chi^2_q.

Why it works

If the excluded variables matter, they should explain part of what the restricted model left in \tilde u, and R^2_{\tilde u} will not be close to zero.

• Only the restricted model is estimated. This is the attraction of LM tests: the same idea returns in the tests for heteroskedasticity (Topic 11).
• The \chi^2 distribution is a large-sample result, like Theorem 5.2: no normality assumption, but n must be large.

Interactive: the LM test step by step

Parents’ education and birth weight (n = 1{,}191):

\begin{aligned} H_0&: \beta_{motheduc} = 0,\;\; \beta_{fatheduc} = 0\\ H_1&: \beta_{motheduc} \neq 0 \;\text{ or }\; \beta_{fatheduc} \neq 0 \end{aligned}

F and LM on the same data

Birth weights again: the F test and the LM test of the same H_0 against the same H_1.

data("bwght")
ur  <- lm(bwght ~ cigs + parity + faminc + motheduc + fatheduc, data = bwght)
r   <- lm(bwght ~ cigs + parity + faminc, data = model.frame(ur))
aux <- lm(resid(r) ~ cigs + parity + faminc + motheduc + fatheduc, data = model.frame(ur))
LM  <- nobs(aux) * summary(aux)$r.squared
round(c(F = anova(r, ur)$F[2],  p_F = anova(r, ur)$`Pr(>F)`[2],
        LM = LM, p_LM = pchisq(LM, df = 2, lower.tail = FALSE)), 4)

#>      F    p_F     LM   p_LM 
#> 1.4373 0.2380 2.8821 0.2367

The finite-sample F test (p = 0.238) and the large-sample LM test (p = 0.237) give the same answer, as asymptotic theory promises.

Asymptotic efficiency

• Under the Gauss–Markov assumptions, OLS is also asymptotically efficient: among a broad class of consistent estimators, it has the smallest asymptotic variance.
• In large samples, the F and LM tests usually lead to the same conclusion, as in the birth weight example (p = 0.238 vs. 0.237).

A researcher has 20,000 observations and strongly skewed errors, but MLR.1–MLR.5 hold. What can she use?

A) Nothing: the t test requires normal errors

B) The usual t and F tests, which are approximately valid by asymptotic normality

C) Only the LM test

With a large sample, Theorem 5.2 makes the t and F statistics approximately standard normal and F, even without MLR.6.

R toolkit for this topic

# 1. Consistency is about what happens as n grows: simulate it
set.seed(1)
sapply(c(10, 100, 1000, 10000), function(n) {
  x <- rnorm(n); y <- 1 + 0.5 * x + rnorm(n)      # true beta1 = 0.5
  coef(lm(y ~ x))[2]
})

# 2. Asymptotic bias does not go away with n: add a correlated omitted variable
#    y <- 1 + 0.5*x1 + 1*x2 + rnorm(n), x2 correlated with x1, then regress y on x1 only

# 3. Large-sample critical values
qt(0.975, df = 1000); qnorm(0.975)

# 4. LM test (large-sample alternative to F): n * R-squared of the auxiliary regression
aux <- lm(resid(restricted) ~ ., data = model.frame(unrestricted))
pchisq(nobs(aux) * summary(aux)$r.squared, df = q, lower.tail = FALSE)

Key takeaways

1. Consistency: as n \to \infty, the sampling distribution of \hat\beta_j collapses on \beta_j.
2. OLS is consistent under MLR.4′ (\text{Cov}(x_j, u) = 0), which is weaker than MLR.4.
3. An omitted variable makes OLS inconsistent: \text{plim}\,\tilde\beta_1 = \beta_1 + \beta_2\delta_1. More data does not help.
4. Asymptotic normality (Theorem 5.2): t, F and LM tests are approximately valid in large samples without MLR.6.
5. Standard errors shrink at the rate 1/\sqrt{n}.
6. OLS is also asymptotically efficient under the Gauss–Markov assumptions.

Next topic

Further issues: units of measurement, quadratics and interactions, adjusted R^2 and prediction.

Wooldridge ch. 6 Heiss ch. 6