The Likelihood Ratio Test

You almost never test an idea in a vacuum. You test it against a rival. Is this strategy's win rate really a coin flip, or is it something else? Does adding two more factors to a model actually explain returns, or does it just soak up noise? In every one of those questions there is a plain story and a richer story, and the richer story has extra freedom to bend itself toward the data. It will always fit at least as well. So "it fits better" proves nothing on its own. You need a way to ask whether it fits enough better.

The tailor analogy

Think of two tailors making you a jacket. One works from a fixed pattern. The other takes your measurements and adjusts as he goes. The second jacket will fit better — that is guaranteed, not evidence of skill. The only interesting question is whether it fits so much better that the extra adjustments were clearly worth making, or whether the improvement is the sort of thing you would get from any tailor allowed to fiddle.

The likelihood ratio test is the statistical version of that judgement. It compares the best fit the plain story can manage against the best fit the flexible story can manage, and it has a fixed exchange rate for how much improvement a single extra adjustment is worth by chance alone.

Putting numbers on "fit"

The measure of fit is likelihood: given a candidate story, how probable was the data you actually observed? Write $L(\theta)$ for that probability, where $\theta$ — the Greek letter theta — stands for whatever the story's adjustable settings are (a win rate, a mean, a set of regression coefficients). A bigger $L$ means the story assigns more probability to what really happened.

The plain story confines $\theta$ to a restricted set; the flexible story lets it roam free. Take the best each can do and divide:

$$\Lambda = \frac{\text{best likelihood the plain story can reach}}{\text{best likelihood the flexible story can reach}} .$$

In words: how much of the flexible story's fit does the plain story already give you? Because the flexible story is never worse, $\Lambda$ sits between 0 and 1. Close to 1 means the restriction cost you almost nothing. Close to 0 means the restriction is strangling the fit.

Working with the raw ratio is awkward, so we take logs and flip the sign:

$$\text{LR} = -2\ln\Lambda .$$

In words: twice the drop in log-likelihood caused by imposing the plain story. Zero means no damage; large means a lot of damage. The picture below is the whole idea — a hill of fit, the peak the flexible story climbs to, and the lower spot the plain story is pinned at.

The exchange rate

How big a gap is suspicious? Wilks' result gives the answer, and it is remarkably tidy: when the plain story is true and the sample is reasonably large, $\text{LR}$ behaves like a draw from a chi-square distribution whose degrees of freedom $d$ equal the number of adjustments you took away.

$$\text{LR} \;\approx\; \chi^2_d, \qquad d = \text{number of restrictions imposed} .$$

In words: each knob you freeze up is worth, on average, about one unit of $\text{LR}$ for free. Anything much beyond that budget is real signal. The cutoffs you need are worth memorising:

Restrictions $d$	5% cutoff	1% cutoff
1	3.84	6.63
2	5.99	9.21
3	7.81	11.34

The first row is a familiar number in disguise: $3.84 = 1.96^2$. Freeze one parameter and the likelihood ratio test is the squared version of the usual two-sided normal test. Drag the shaded band in the explorer below out to $\pm 1.96$ and watch it capture 95% of the curve — that is the same 5% you are spending when you use the cutoff 3.84.

Worked example one: is this win rate really a coin flip?

A strategy took 100 independent trades and won 60. Plain story: the true win rate is $p = 0.5$. Flexible story: $p$ is whatever it wants to be.

Step 1 — score the plain story. For 60 wins and 40 losses at rate $p$, the log-likelihood (ignoring a constant that cancels in the ratio) is $60\ln p + 40\ln(1-p)$. At $p = 0.5$:

$$60\ln(0.5) + 40\ln(0.5) = 100 \times (-0.6931) = -69.31 .$$

Step 2 — score the flexible story. Its best setting is just the observed rate, $\hat p = 0.60$:

$$60\ln(0.6) + 40\ln(0.4) = -30.65 + (-36.65) = -67.30 .$$

Step 3 — take the gap and double it.

$$\text{LR} = 2\big(-67.30 - (-69.31)\big) = 2 \times 2.01 = 4.03 .$$

Step 4 — compare. One parameter was frozen, so $d = 1$ and the cutoff is 3.84. Since $4.03 > 3.84$, you reject the coin-flip story at the 5% level — but only just. The p-value is about 0.045. Sixty wins out of a hundred is barely enough to distinguish an edge from luck, which is a useful thing to feel in your bones before you size a position on 100 trades.

Worked example two: do two extra factors earn their keep?

You regress 120 months of fund returns on the market alone, then on the market plus size and value. Leftover squared error falls from 0.0500 to 0.0470. With normally distributed errors the likelihood ratio collapses to a comparison of leftover error:

$$\text{LR} = n \ln\!\left(\frac{\text{leftover error, plain}}{\text{leftover error, flexible}}\right).$$

In words: the statistic depends only on the proportional shrinkage in unexplained variation, scaled by sample size. Plugging in:

$$\text{LR} = 120 \times \ln\!\left(\frac{0.0500}{0.0470}\right) = 120 \times \ln(1.0638) = 120 \times 0.0619 = 7.43 .$$

Two coefficients were added, so $d = 2$ and the cutoff is 5.99. Since $7.43 > 5.99$, the two factors are pulling real weight; the p-value is about 0.024. Notice how small the raw improvement looked — error fell by 6% — and how the test still called it. Sample size is doing that work: with only 30 months the same 6% shrinkage would give $\text{LR} = 1.86$, comfortably inside the noise budget.

What this means in practice

Most of the tests a quant runs are secretly this test. The F-test comparing nested regressions, the check that a GARCH model needs a second lag, the test that a fitted tail parameter differs from the Gaussian value — all likelihood ratios wearing different clothes. Knowing that means you can build a test for a situation no textbook covers: write down both likelihoods, maximise each, double the gap, count restrictions.

It also disciplines model selection. Adding features to a return model always lifts in-sample fit; the likelihood ratio gives you a price list for that lift, so you stop being impressed by improvements any extra parameter would have delivered. When you have many candidate models rather than two nested ones, the same logic reappears as Data-Snooping Bias control.

The practical failure mode is the approximation itself. Wilks' chi-square result is a large-sample promise. With a few dozen observations, or when the tested value sits on the edge of what is allowed (testing whether a variance is zero, when variances cannot go negative), the reference curve is wrong and your p-value is fiction. Simulate the null yourself with Bootstrap and Resampling instead of trusting the table.

The likelihood ratio, the The Wald Test and the score test all approximate one another in large samples and often disagree in small ones; when they do, the likelihood ratio is usually the one to trust, because it compares actual fitted models rather than local curvature. Its optimality for simple two-way comparisons comes from the The Neyman-Pearson Lemma, and its reference curve is the The Chi-Squared Distribution.