# How is the ARIMA model a valid approach for forecasting Economic Variables?

Ive noticed the use of univariate forecasting methods (i.e the ARIMA model) employed in forecasting inflation (see references below).

How is this a valid approach in terms of forecasting economic variables? does this approach not omit the large amount of economic theory which determines how these variables are determined (i.e via equilibrium and interaction of variables other than themselves?)

1.https://scholar.harvard.edu/files/stock/files/forecastinginflation.pdf see pages 6-7

So your question is basically asking why a purely statistical model would be better at forecasting than a model based on theoretical relationships. For example, think about a simple form of the ARIMA model i.e. an AR model where you're just using lags of the process to explain the process itself. Now think about what is contained in the lags of the process... If the true process (i.e. the DGP) is determined by theoretical relationships between other variables then these will automatically be included in the lag. So the ARIMA model is a statistical way of modeling the process to make sure you capture the statistical properties of the underlying relationship but does not necessarily provide an explanation for what is going on. In forecasting, especially when series are volatile and erratic, you often find that simple statistical models such as these do a better job since the underlying relationships may not be very stable or may be more complicated than any theory we can come up with. This is related to a recent article which you may find interesting: http://www.sciencedirect.com/science/article/pii/S0169207017300997

• Wow there is a lot of smack talk in this paper. – EconJohn Nov 2 '17 at 15:58

In short, for the purposes of forecasting, it is often convenient and even beneficial to ignore economic theory. The restrictions implied by economic theory is necessary for inference when the question of interest is causal in nature (e.g., when you want to understand the underlying causal relationships). However, if the forecast you're interested is more of the variety of a sophisticated extrapolation, then something simple like ARIMA may be more helpful and more accurate.

I think the following are helpful references.

## Why would I not be interested in the underlying causal relationship?

Consider the following example from "Introduction to Econometrics" (3rd Edition), by Stock and Watson (p. 517).

The simplest regression model in Chapter 3 related test scores to the student-teacher ration (STR): $$\widehat {Test Score} = 989.9 - 2.28 \times STR \tag{14.1}$$ As was discussed in Chapter 6, a school superintendent, contemplating hiring more teachers to reduce class sizes, would not consider this equation to be very helpful. The estimated slope coefficient in Equation (14.1) fails to provide a useful estimate of the causal effect on test scores of the student-teacher ratio because of probable omitted variable bias arising from omission of school and student characteristics that are determinant of test scores and that are correlated with the student-teacher ratio.

In contrast, as was discussed in Chapter 9, a parent who is considering moving to a school district might find Equation (14.1) more helpful. Even though the coefficient does not have a causal interpretation, the regression could help the parent forecast test scores in a district for which they are not publicly available. More generally, a regression model can be useful for forecasting even if none of its coefficients has a causal interpretation. From the perspective of forecasting, what is important is that the model provides as accurate a forecast as possible.

Along these same lines, this is a point that is brought up frequently when discussing Machine Learning. Machine Learning may not be so useful in helping to understand underlying causal relationships. However, often we are more interested in the predictions/forecasts. Sendhil Mullainathan in this talk on "Machine Learning and Prediction in Economics and Finance" gives the useful "rule of thumb" rubric---are you more interested in the $\hat y$ or the $\hat beta$?

## Why would something like ARIMA forecast better than a model that better reflected economic theory?

Consider this useful passage from "A Guide to Econometrics" (6th Edition) by Peter Kennedy (p. 333).

The main competitors to econometric models for forecasting purposes are Box-Jenkins, or ARIMA (autoregressive integrated moving average), models explained in some detail in chapter 19. Univariate Box-Jenkins models are sophisticated extrapolation methods, using only past values of the variable being forecast to generate forecasts; they ignore the many explanatory variables that form the foundation of econometric models. There are several reasons why forecasters should be interested in these naive models: thanks to improved computer software, they are easy and cheap to produce; the extra information required to estimate a proper econometric model may be expen­sive to obtain ; forecasts from such models can serve as a useful benchmark for com­parison purposes; forecasts from this process can be combined with other forecasts to produce improved forecasts ; and they are useful as a preliminary step for further modeling---they clarify the nature of the data and make clear what behavior patterns require explanation.

During the 1970s controversy raged over the relative forecasting merits of econo­metric models and ARIMA models, prompted by studies claiming the superiority of the ARIMA models. As noted in chapter 19, this led to a synthesis of the two approaches, and prompted the development of models, such as error-correction models (ECMs), which paid more attention to dynamics. In retro spect, the reason why econometric models performed so poorly in these comparisons was because of misspecification errors in the econometric models, primarily with respect to their dynamic structure. It is generally acknowledged that whenever specification or conditioning errors ren­der econometric models impractical (which some claim is most of the time), the Box-Jenkins approach has considerable merit for forecasting. It is also recognized that if an econometric model is outperformed by an ARIMA model, this is evidence that the econometric model is misspecified.

## Can a univariate model like ARIMA represent a rational expectations equilibrium?

Not really. Most economics theory employ rational expectations that lead to a nonlinear system of equations (in multiple variables) that describe the model's dynamics. Often, these are approximated (e.g., log-linear approximation), so that they can be solved and estimated using something like a linear state space model---but many still require multiple variables (here, arranged in a vector) to describe the full dimensionality of the model. For an example of one particular model that uses rational expectations, see p. 30 of "Recursive Models of Dynamic Linear Economics" by Hansen and Sargent.

Consider a stochastic process $\{p_t\}$ related to a stochastic process $\{m_t\}$ via $$p_t = \lambda E_t p_{t+1} + \gamma m_t \tag{(2.4.38)}$$ where $$m_t = G x_t \tag{(2.4.39)}$$ and $x_t$ is governed by $$x_{t+1} = A x_t + C w_{t+1}, \text{ for } t = 0,1,2,...$$

... Collecting results, we have that $(p_t, m_t)$ satisfies \begin{align*} \begin{bmatrix}{p_t \\ m_t }\end{bmatrix} &= \begin{bmatrix} \gamma G(I - \lambda A)^{-1} \\ G \end{bmatrix} x_t \\ x_{t+1} &= A x_t + C w_{t+1}. \end{align*} [This system] embodies the cross-equation restrictions associated with rational expectations models: note that the same parameters in $A,G$ that pin > down the stochastic process for $m_t$ also enter the equation that determines > $p_t$ as a function of the state $x_t$.

This equation, written flexibly, allows for many state variables and many (possibly orthogonal) shocks. As such, this cannot be represented by a simple univariate model like ARIMA.

Many other examples can be found in "Structural Econometrics" by Dejong and Dave or more in "Recursive Models of Dynamic Linear Economics", by Hansen and Sargent. Some will only have a single dimensional state space. Some will have much larger state spaces.

However, again, describing the model in full may not be necessary if you're only interested in simple forecasting. Hence, something like ARIMA is an attractive option.

• I'll read this tonight. looks very comprehensive – EconJohn Feb 26 '18 at 21:57
• It would seem that the source to support the argument that AR(p) models are not a result of RE equilibrium is a refutation for your own position. See the structural law of motion at the end of lectures.quantecon.org/py/rational_expectations.html – EconJohn Feb 26 '18 at 23:46
• @EconJohn This doesn't refute it. The law of motion, the one for aggregate productivity, that you point out simply has a single shock and single state variable. This could easily be extend to include more states and more shocks. I have edited my answer to include an example that more clearly illustrates this. Most models you see in modern macro textbooks are RE models. However, most of them don't explicitly state that they use RE. The examples I've chosen are those that both say that they are RE and exhibit (or could exhibit) multiple irreducible dimensions. Hope this helps! – jmbejara Feb 27 '18 at 0:44

I came across the answer to the validity of univariate modelling in economics (albet indirectly) from the book called A Rational Expectations Approach to Macroeconometrics. This answer is based on the material in the book, for a better understanding I reccomend actually going through it.

In short the validity of ARIMA models is based on the assumption that the market is always in equilibrium or efficient1.

In order to understand how use of ARIMA models in Econometrics is a valid way to forecast economic variables, we must understand models which considers the case of for rational expectations (other wise referred to as efficient markets).

your standard efficient markets model is:

$$y_t=\tilde{y}_t+\sum_{i=0}^N\beta_i(X_{t-i}-X_{t-i}^e)+\epsilon_t$$ where:

• $y_t=$ Economic variable of interest at time $t$ like unemployment or real output.
• $\tilde{y}_t=$Natural rate or equilibrium of economic variable at time $t$
• $\beta_i=$ coefficents.
• $X_{t-i}=$ an aggreagte demand variable like money growth, Inflation or Nominal GDP growth.
• $X^e_{t-i}=$ Anticipated aggregate demand conditional on knowing all information at point $t-i$
• $\epsilon_t=$ an error term

This model accounts for variability that can be present when a market does not have all information pertaining to $y_t$.

combining this formula using a specification used with the Lucas supply function: $$\tilde{y}=\sum_{i=1}^N\lambda_iy_{t-i}$$

combining the two equations we get:

$$y_t=\sum_{i=1}^N\lambda_iy_{t-i}+\sum_{i=0}^N\beta_i(X_{t-i}-X_{t-i}^e)+\epsilon_t$$

This model tells us that if rational expectations hold,the relationship of $y_t=\sum_{i=1}^N\lambda_iy_{t-i}$, thus providing validity for the use of uni variate models in economics.

For more reading on this topic, below is posted links to the three chapters which helped me:
1)http://www.nber.org/chapters/c10241.pdf
2)http://www.nber.org/chapters/c10243.pdf
3)http://www.nber.org/chapters/c10244.pdf

UPDATE: Upon Contacting Dr. Paul Middleditch on Twitter he says that the RHS variables must be replaced with a proxy which is normally the value of the previous period. ARIMA methodology satisfies such constraints.2

1. Mathematically this statement can be written as: $$\mathbb{E}_m(X_t|\phi_{t-1})=\mathbb{E}(X_t|\phi_{t-1})$$

where the expected value given by the market of $X_t$ conditional on $\phi_{t-1}$ the set of all available information at time $t-1$, is equal to the true expected value of $X_t$

It follows logically that $\mathbb{E}[X_t-\mathbb{E}_m(X_t)|\phi_{t-1}]=0$

2. a Youtube series he prodouced on rational expectations
• (3) You write that $y_t=\tilde{y}_t+\sum_{i=0}^N\beta_i(X_{t-i}-X_{t-i}^e)+\epsilon_t$ is your efficient markets model. This isn't the case. Even in the book you references, this is references as "a model that satisfies the efficient markets condition." This itself is even a little out of date, but that's besides the point. I would be careful to not claim that rational expectations is more than it is---that is, that economic actors are endowed with beliefs about the underlying statistical model that match the true model. – jmbejara Feb 23 '18 at 19:31
• (4) Also, the model that is used in the book is not necessarily univariate. Market expectations in $X^e_{\tau}$ may be and probably are formed on other variables besides $X$. See p. 15 of the cited book. – jmbejara Feb 23 '18 at 19:32