Usually when creating a regression model using cross sectional data we assume a dependent and independent variable.

but in AR,MA, ARMA and ARIMA models forecast based on the previous periods values or disturbance terms.

What is the rationale for univariate time series modelling and why would one prefer to use a univariate forecast as opposed to a multiple variable one?

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    $\begingroup$ When to prefer a univariate forecast to a bivariate one? For example, when you only have data on one variable. $\endgroup$ – Richard Hardy Oct 4 '17 at 9:31
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    $\begingroup$ In what sense are you using "univariate" and "bivariate" in this question? I have to ask, because as @Richard hints, in most applications the distinction is glaringly obvious: either you are forecasting one variable per time or multiple variables per time--and that's the distinction usually understood by these terms. $\endgroup$ – whuber Oct 4 '17 at 13:38
  • $\begingroup$ @RichardHardy so its just preferred in absence of other data? $\endgroup$ – EconJohn Oct 4 '17 at 15:35
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    $\begingroup$ It would help to mention this explicitly. Since, in its current version, this post refers explicitly only to univariate models, it's potentially confusing. It's confusing, too, because of the obvious answer: you use multivariate forecasts when you have multiple values to forecast (at each time). It would help to show why this might not be a complete or satisfying answer in your case. $\endgroup$ – whuber Oct 4 '17 at 18:54
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    $\begingroup$ Here the term multivariate is not appropriate ... the term to be used is multiple ... in fact you are referring to simple AR vs ADL that are so called Autoregrssive distributed lag models where you have lags of additional predictors in addition to lags of the dependent variable $\endgroup$ – gioxc88 Oct 27 '17 at 4:07

You have two distinct questions in your post:

  • Why use univariate models? and
  • Why use forecasts from univariate models?

I will give several answers that fit both of the above and then another answer that only addresses the former.

A forecast based on a univariate time series may be preferred over a forecast based on a multivariate time series in the following cases:

  1. You only observe one time series $y_t$. Here you do not have a choice. Clearly, only a univariate model can deliver a feasible forecast.
  2. You observe a multivariate time series $(y_t,X_t)$ where $X_t$ may be univariate or multivariate, but $X_t$ do not have any influence on $y_t$. Including $X_t$ in the model only introduces noise in your forecasts and hence makes them less accurate.
  3. You observe a multivariate time series $(y_t,X_t)$ where $X_t$ may be univariate or multivariate, and $X_t$ does have some influence on $y_t$. However, the influence is too small to be measured and/or modelled accurately. Then including $X_t$ in the model introduces more noise than signal, leading to poorer forecasts.

    This is a matter of signal to noise ratio and the sample size. The smaller the ratio and the sample size, the less grounds there are for choosing a multivariate model over a univariate one. This is also what Aksakal talks about in his answer.
  4. Forecasting needs to be done fast. Multivariate models generally take longer to estimate than their univariate counterparts due to the increased complexity. Speed can be important when you have to react to new information very quickly (such as in some forms of trading on the financial markets) or need to process large amounts of data (thousands or millions of time series).
  5. The multivariate counterpart of the univariate model is too complex to estimate. This may be true for ARIMA models when the dimension of $X_t$ is relatively high. Univariate ARIMA models are moderately hard to estimate, but multivariate ones have a notoriously high apetite for computational resources. Thus sometimes estimating a multivariate model and producing forecasts from it is infeasible while there are no problems with the univariate model and its forecasts.

Regarding modelling for other purposes than forecasting, all of the above still holds true by substituting "forecasts" with "coefficient estimates". Moreover,

  1. You may be interested in projecting the behaviour of a time series only on its own past but not on any other series. Thus you are not interested in causal modelling or representing the true relationships among all the variables available to you but rather in visualizing how a single series develops in its own historical perspective. Then a univariate model is preferable.

P.S. Most of the discussion above applies not only in the time series setting but also more generally.

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  • $\begingroup$ This should be in a introductory textbook on time series analysis when introducing univariate modelling. amazing $\endgroup$ – EconJohn Oct 29 '17 at 16:21

If you do not have a lot of data, then univariate series could be the only way to go. For instance, in the simplest 2D VAR(1) you have the following: $$X_t=\beta_0+\Phi_1X_{t-1}+\varepsilon\\\varepsilon\sim\mathcal{N}(0,\Sigma)$$ where $X$ - two dimensional vector of observable series, $\Phi_1$ - 2x2 matrix of lags, and $\Sigma$ - 2x2 matrix of error covariances. So, you have 2 intercepts, 4 lag coefficients, 2 variances and one covariance to estimate, i.e. 9 parameters in total.

If this was two univariate models you'd have: one intercept, one autocorrelation coefficient and one variance. This makes it 6 parameters in total, i.e. 3 fewer parameters to estimate. In general the number of parameters is linear to the number of dependent variables, while in VAR it's quadratic growth. If you have multiple lags and many dependent variables the number of parameters quickly gets out of control.

When you increase number of parameters the system becomes unstable, very sensitive to small variations in the sample, unless your sample is very large.

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