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I have identified a model using principal component regression where $Y_t$ is explained by 4 factors such as:

$$Y_t = \beta_1 X_{1t} + \beta_2 X_{2t} + \beta_3 X_{3t} + \beta_4 X_{4t} + \epsilon_t$$

Where: $Y, X_1, X_2, X_3, X_4$ are $\text{I(1)}$ variables.

If I check that $\epsilon_t$ is stationary, can I assume that the variables are co-integrated and hence there was no spurious regression in my principal component regression?

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  • $\begingroup$ As this is a question about statistics/econometrics rather than finance, it should be moved to Cross Validated. $\endgroup$ – Richard Hardy Aug 29 '16 at 16:33
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If a regression of an integrated variable on one or more integrated variables yields a stationary residual, the variables are cointegrated. This is a special case of the definition of cointegration.

Make sure to use appropriate null distribution and corresponding critical values when testing for absence of a unit root in $\epsilon_t$. Since $\epsilon_t$ is a residual rather than raw data, it is more likely to appear stationary by construction; that is why you need critical values as in Engle & Granger's procedure rather than the regular critical values of an augmented Dickey-Fuller (ADF) test.

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