How do you judge whether an OLS coefficient is significant from its estimate and standard error?

## The ratio A regression output table for each coefficient prints four numbers: the estimate, its standard error, a t-statistic and a p-value. The last two are computed from the first two, so given any two of them you can reconstruct the rest. The t-statistic is `t = (estimate - hypothesised value) / SE(estimate)` and the hypothesised value defaults to zero, because the question the table is answering is *does this predictor carry any linear association with the outcome once the others are held fixed*. So in practice `t = estimate / SE`. Read literally, it counts how many of the coefficient's own standard errors separate the estimate from zero. ## Degrees of freedom The reference distribution is a t distribution with `n - p` degrees of freedom, where `n` is the number of observations and `p` is the number of estimated coefficients **including the intercept**. Simple regression with one predictor uses `n - 2`. A model with four predictors and an intercept fit on 200 rows uses `195`. Forgetting the intercept in that count is a routine slip and matters only when `n` is small. Why `n - p`? The standard error is built from `s = sqrt(SSE / (n - p))`, and `p` residual directions have been used up fitting the coefficients themselves. Because `s` is estimated rather than known, the reference distribution has heavier tails than a normal, and the heaviness depends on how much data was left over. ## Reading the bar The two-sided p-value is the probability, computed under the model and assuming the true coefficient is zero, of drawing a t-statistic at least as extreme in absolute value as the one observed. Small p-values mean the observed ratio would be unusual if the coefficient really were zero. The practical shortcut every analyst carries is the **two-sigma bar**: `|t| >= 2` is roughly the 5% cutoff. The exact critical values show how good the approximation is — about 2.05 at 28 degrees of freedom, converging down toward 1.96 as degrees of freedom grow, but climbing to about 2.26 at 9 degrees of freedom and higher still below that. So with a healthy sample, eyeballing `estimate / SE` against 2 is reliable; with a dozen rows it is optimistic and you should look up the actual critical value. ## The interval says the same thing The confidence interval for the coefficient is `estimate +/- (critical value) * SE`. Zero falls inside that interval precisely when `|t|` is below the critical value. So *the interval excludes zero* and *the coefficient is significant at that level* are the same statement, arrived at from the same two numbers. Given a printed estimate and standard error you can produce both without any further computation — a common whiteboard ask. ## Worked reconstruction An estimate of `0.36` with a standard error of `0.30` gives `t = 1.2`. That is well short of 2, so the coefficient does not clear the usual bar, and the approximate 95% interval `0.36 +/- 2 * 0.30 = [-0.24, 0.96]` straddles zero, agreeing. An estimate of `0.36` with a standard error of `0.09` gives `t = 4.0` and an interval of roughly `[0.18, 0.54]`, which clears it comfortably. Identical point estimates, opposite conclusions — the standard error is doing all the work. ## Testing against a value other than zero Sometimes zero is not the interesting null. If a pricing model's slope should be `1` under a proposed pass-through assumption, test `t = (estimate - 1) / SE` with the same degrees of freedom. Interviewers use this to check whether a candidate has memorised the output column or understands the construction. ## What to say out loud Name the ratio, name the reference distribution and its degrees of freedom, give the roughly-2 shortcut with the caveat for small samples, and note that the interval-excludes-zero check is the same test. That is the complete answer at this level.