{"schemaVersion":2,"courseId":"iv2sls","module":{"id":"iv2sls-why-ols-fails","title":"1 · Why OLS fails","summary":"Endogeneity from omitted variables and simultaneity makes OLS biased and inconsistent.","stages":[{"type":"read","title":"The exogeneity assumption","html":"

What OLS quietly assumes

Write the model as

y = β₀ + β₁·x + ε

OLS gives a consistent, unbiased estimate of β₁ only if the regressor is exogenous: the error term ε must be uncorrelated with x, written Cov(x, ε) = 0. Intuitively, everything we left out of the model (bundled into ε) must not move together with x.

When instead Cov(x, ε) ≠ 0 we say x is endogenous. OLS then attributes to x variation that really belongs to ε, and β̂₁ is pulled away from the truth.

Key idea. OLS regresses y on all the comovement between x and y. If part of that comovement runs through ε, OLS cannot tell the causal part from the contaminated part — and the bias does not shrink as n → ∞.

Two classic ways endogeneity arises

","id":"iv2sls-why-ols-fails-01","skillIds":["iv2sls.endogeneity"],"estimatedMinutes":6,"difficulty":"applied","prerequisiteStageIds":[]},{"type":"read","title":"The sign and size of the bias","html":"

How big is the OLS bias?

Suppose the true structural model has a confounder u that we cannot observe:

x = π·u + v, y = β₀ + β₁·x + γ·u + η

Because u appears in both equations, x carries information about u, and u sits inside the regression error ε = γ·u + η. The asymptotic OLS bias is

plim β̂₁ = β₁ + γ · Cov(x,u)/Var(x)

So the bias has the sign of γ·π (the product of how u affects y and how u affects x). If both effects are positive, OLS overstates β₁; if they have opposite signs it understates it.

Key idea. More data sharpens a wrong answer: the standard error shrinks around a biased point. Precision is not the same as correctness.

This is why we need a new strategy. We cannot fix endogeneity by collecting more rows — we need extra information in the form of an instrument, the subject of Module 2.

","id":"iv2sls-why-ols-fails-02","skillIds":["iv2sls.instrument-relevance"],"estimatedMinutes":6,"difficulty":"applied","prerequisiteStageIds":["iv2sls-why-ols-fails-01"]},{"type":"code","title":"Watch OLS miss the truth","note":"True β₁ = 2. A hidden confounder u raises both x and y, so OLS overshoots — and tightening n does not help.","code":"import numpy as np, statsmodels.api as sm, matplotlib.pyplot as plt\nrng = np.random.default_rng(0)\n\ndef ols_slope(n):\n u = rng.normal(0,1,n) # hidden confounder\n x = 1.0*u + rng.normal(0,1,n) # u drives x\n y = 1 + 2*x + 3*u + rng.normal(0,1,n)# u also drives y (true beta1=2)\n return sm.OLS(y, sm.add_constant(x)).fit().params[1]\n\nns = [50,100,200,500,1000,2000]\nest = [ols_slope(n) for n in ns]\nfor n,b in zip(ns,est):\n print(f'n={n:5d} OLS beta1_hat = {b:.3f} (true = 2.000)')\n\nplt.figure(figsize=(6,3.7))\nplt.axhline(2,color='#c9c6ac',ls='--',lw=2,label='true beta1 = 2')\nplt.plot(ns,est,'o-',color='#da9100',lw=2,label='OLS estimate')\nplt.xlabel('sample size n'); plt.ylabel('beta1_hat')\nplt.title(f'OLS converges to {est[-1]:.2f}, not 2 (inconsistent)')\nplt.legend()","id":"iv2sls-why-ols-fails-03","skillIds":["iv2sls.endogeneity"],"estimatedMinutes":10,"difficulty":"applied","prerequisiteStageIds":["iv2sls-why-ols-fails-02"]},{"type":"interactive","title":"Confounding slider","note":"γ controls how strongly the hidden u also pushes y. At γ = 0 there is no confounding and OLS is fine; crank it up and the estimate drifts away from the true 2.","params":[{"name":"gamma","label":"Confounder strength γ (effect of u on y)","min":-4,"max":4,"step":0.25,"value":3}],"template":"import numpy as np, statsmodels.api as sm, matplotlib.pyplot as plt\nrng = np.random.default_rng(0); n=800\nu = rng.normal(0,1,n)\nx = 1.0*u + rng.normal(0,1,n)\ny = 1 + 2*x + {{gamma}}*u + rng.normal(0,1,n)\nres = sm.OLS(y, sm.add_constant(x)).fit()\nb = res.params[1]\nplt.figure(figsize=(6,3.7))\nplt.scatter(x,y,s=10,color='#da9100',alpha=.4)\nxs=np.linspace(x.min(),x.max(),50)\nplt.plot(xs,res.params[0]+b*xs,color='#c9c6ac',lw=2,label='OLS fit')\nplt.plot(xs,res.params[0]+2*xs,color='#af983f',lw=2,ls='--',label='true slope=2')\nplt.legend(); plt.title(f'gamma={{gamma}} -> OLS beta1_hat = {b:.2f} (bias {b-2:+.2f})')","id":"iv2sls-why-ols-fails-04","skillIds":["iv2sls.instrument-relevance"],"estimatedMinutes":8,"difficulty":"applied","prerequisiteStageIds":["iv2sls-why-ols-fails-03"]},{"type":"quiz","title":"Check: source of the bias","prompt":"An unobserved variable raises both schooling x and wages y. We regress wages on schooling by OLS. What happens as the sample grows very large?","choices":["The estimate converges to the true return to schooling","The estimate stays biased; only its standard error shrinks","The bias grows without bound","OLS becomes unbiased once n > 1000"],"answer":1,"hint":"Endogeneity is about consistency, not sampling noise.","explain":"Omitted-variable bias does not vanish with n. The point estimate converges to a wrong number while its SE shrinks around that wrong number — precise but inconsistent.","id":"iv2sls-why-ols-fails-05","skillIds":["iv2sls.endogeneity"],"estimatedMinutes":5,"difficulty":"applied","prerequisiteStageIds":["iv2sls-why-ols-fails-04"],"variantId":"iv2sls-why-ols-fails-05-core"},{"type":"quiz","title":"Check: sign of the bias","prompt":"In x = π·u + v and y = β₁·x + γ·u + η, suppose γ > 0 and π > 0. OLS will tend to…","choices":["overstate β₁","understate β₁","estimate β₁ exactly","give a negative β̂₁ regardless"],"answer":0,"hint":"The asymptotic bias has the sign of γ·π.","explain":"With γ·π > 0 the omitted u creates positive comovement between x and the error, so OLS attributes extra positive slope to x and overstates β₁.","id":"iv2sls-why-ols-fails-06","skillIds":["iv2sls.instrument-relevance"],"estimatedMinutes":5,"difficulty":"applied","prerequisiteStageIds":["iv2sls-why-ols-fails-05"],"variantId":"iv2sls-why-ols-fails-06-core"}]}}