# How to show that any Gaussian time-series is linear one?

In this paper I saw the following statement:

If the time series is Gaussian (i.e., normally distributed) then the best linear forecast is in fact the best of all possible forecasts: No nonlinear forecast can do better in terms of mean squared prediction error. Thus, as long as the series is Gaussian, we need look no further than the linear methods (e.g., ARMA forecasting) already presented.

I think it is a very strong, general and interesting statement. Is it easy to prove, that a time-series whose values are normally distributed are given ("generated") by a linear model? Or, alternatively, any nonlinear time-series should have a non Gaussian distribution of values.

• My takeaway from @Alecos answer is that the statement is not generally true. It is true only in the particular case where the data generating process is linear. – Richard Hardy Dec 2 '15 at 20:27

This is more general than the normal case.

If we take as criterion of optimality the minimization of Mean Squared Forecasting Error (and it is not the only one, but perhaps it is the more widely used), then the best predictor is the Conditional Expectation (i.e. the conditional expected value treated as a function), for any distributional assumption. There are some threads here in CV that contain this proof, for example, https://stats.stackexchange.com/a/71869/28746

Then, for those joint distributions that result in Conditional Expected Value being a linear function of the conditioning variables, it follows that the best in minimum MSFE sense predictor will be a linear function of the data.

Linear Conditional Expectations characterize the symmetric elliptical class as well as many of the Pearson family of multivariate distributions. The normal distribution belongs here.

• I think it is the second paragraph of your answer that is crucial for answering the question in the OP. If I understand it correctly, the claim in the OP is wrong because not all joint distributions result in Conditional Expected Value being a linear function of the conditioning variables. Is that right? – Richard Hardy Dec 2 '15 at 19:23
• @RichardHardy Indeed, many joint distributions do not result in linear CEFs (eg. exponential, logistic, log-normal). – Alecos Papadopoulos Dec 2 '15 at 19:38
• I am a little confused now. In a "for dummies" fashion, is the OP right or wrong? It considers the Gausian case in particular. Does there exist a case where the dependent variable is Gaussian for which a nonlinear forecast fares better than all possible linear ones in terms of MFSE? – Richard Hardy Dec 2 '15 at 19:44
• @RichardHardy It may certainly be the case, if the predictors (be it lags of the dependent variable and/or other variables) do not have a joint distribution with the dependent variable that gives a linear CEF. In such a case, the CEF remains best in MSFE terms, but the linear model misrepresents the true CEF, and so incorporates a misspecification error. It can be shown that least-squares estimation of a linear form estimates consistently the first-order Taylor expansion of the true non-linear CEF evaluated at the true expected values of the predictors. – Alecos Papadopoulos Dec 2 '15 at 20:02
• Sorry, I still fail to understand. First you say that "it may certainly be the case" but then seemingly contradict yourself (which is most likely my false understanding) saying that "the CEF remains best in MSFE terms". – Richard Hardy Dec 2 '15 at 20:11

In my opinion, there are two different questions being asked here, and the answer to the first is a Yes.

Given that a time series is Gaussian (meaning that all choices of $n \geq 1$ random variables $X_{t_1}, X_{t_2}, \cdots, X_{t_n}$ from the processes are jointly Gaussian random variables), is it true that the minimum mean-square-error (MMSE) predictor of $X_N$ in terms of any other variables from the series is a linear (or affine) function of the known variables?

As Alecos's answer points out, this is a property enjoyed by jointly Gaussian random variables. It is a very general result (applicable to all random variables, not just Gaussian or jointly Gaussian ones) that the MMSE predictor is the conditional mean of $X_N$ given the other random variables. A simpler predictor of $X_N$ is a linear combination (technically an affine function) of the known variables, and if this function has the smallest mean-square error among all possible linear predictors, then it is called the Linear MMSE (LMMSE) predictor of $X_N$. For jointly Gaussian random variables, the conditional mean (i.e. MMSE predictor) is a linear function of the known variables, and so is also the LMMSE predictor. Thus, the methods (already known to the OP) that produce LMMSE estimators cannot be improved upon for Gaussian time series: those LMMSE estimators are in fact the MMSE estimators, and so we need not have any niggling doubts that had we put in more hard work, we might have been able to come up with a nonlinear predictor that is better than the "easily-found" LMMSE estimator even if the said nonlinear predictor is not as good as the MMSE estimator.

It is worthwhile noting that while most other random variables do not enjoy the property that the LLMSE estimator and the MMSE estimator coincide, it is by no means true that jointly Gaussian random variables are unique in this respect. Consider, for example $(X,Y)$ being uniformly distributed on interior of the triangle with vertices $(0,0), (1,1), (1,0)$. The conditional density of $Y$ given $X=x$ is $U(0,x)$ and thus has mean $\frac x2$. Thus, the MMSE estimator of $Y$ given $X$ is $\frac X2$ which is a linear function of $X$ and hence must be the same as the LMMSE estimator of $Y$ given $X$.

The other question being asked seems to be

Are all Gaussian time series generated by a "linear model"?
or
Are all time series generated by a "linear model" Gaussian time series?
or
Gaussian series cannot be generated by nonlinear models

If by linear model is meant that each random variable $X_N$ in the series can be expressed as

$$X_N = a_N + \sum_{i} b_{i,N} Y_i$$ where the $a$'s and $b$'s are constants and $\{Y_i\}$ is another time series, then the answer to the middle question is No unless of course $\{Y_i\}$ itself is a Gaussian time series. Indeed, since any collection of $L$ jointly Gaussian random variables is obtained via a linear transformation of $M \leq L$ independent standard normal random variables, the answer to the first question is Yes. On the third question, I maintain a discreet silence.