ARMA models, multiple components and recoveries of consumption series from COVID

Introduction – ARMA models and behavior of multiple components

Time series models of the typical ARMA family should be viewed as a tool to tell ourselves stories about how time series behave. They focus us on the question of primary importance - how will the shocks be propagated into future and hence influence future values – and allow us to answer it in simple terms. For example autoregressive model with high coefficient tells us that effect of shocks will be eliminated only gradually and hence propagated over long period of time. On the other hand moving average models of low order tells us that the effect of shocks will disappear quickly. Or finally, moving average model with negative coefficient tells us that the recent shocks will have opposite effects in near future.

This simple pallet of models is very flexible and allows us to capture most of the behavior of real world series. One particular place where it can be applied is in thinking about detailed consumption and production series influenced by the pandemic (think purchases of cars or box office sales). In principle, these series can be thought to be composed of at least 4 components, 3 relating to lockdowns (or more broadly social distancing) and one to the overall economic shock.

The three components related to lockdowns effect capturing the effect of lockdowns which prevent consumption/production; effect of pent-up demand after lockdowns are lifted (i.e. most people who very planning to purchase car will still do so sometime after lockdowns); and effect of substitution, as money saved on some regular expenditures due to lockdowns are spend on some other items due to impossibility of time-substitution (e.g. people will not catch up with all their regular cinema and restaurant visits they have missed). The other component is component capturing effects on confidence and/or expectations about current/future incomes, i.e. the usual component we see in recessions, albeit this time the bad news arrived in more time-compressed manner.

These components behave very differently from each other. Moreover, not all series contain all components, and even if they do, these components might behave differently for different series (think of more persistent lockdowns/social distancing effects for most sensitive industries). This means that in contrast to normal recessions there will be much bigger heterogeneity across individual consumption series. Moreover, the first three are very unique to our current situation and their novelty together with their complexity means that we can easily get confused by temporary movements. It should be therefore worthwhile to consider the different options we might encounter over the next few months and what it means for behavior for given series. Below are few examples.

Example 1 – Spending on restaurants: Pure lockdown shock with no pent-up demand and substitution

The first example applies to series which is affected by the lockdowns, but does not benefit from the pent-up demand and substation effects.  An example could be spending in restaurants – restaurants in many countries were closed for most of April and to a lesser degree May and June, so that the spending in restaurants is substantially lower in these months. At the same time, people will return to restaurants as soon as their re-open, so that July should be back to normal level of spending (or close to that). The graph below translates this discussion into deviations from normal level of spending (left chart) and into growth rate of the series (right chart). The magnitudes are illustrative and not meant to be representative.

Turning to the way this is captured in the standard time series ARMA models, the profile in the chart is achieved by using MA(2) process with single (negative) shock in April. The moving average model extends the effect of shocks for the number of periods corresponding to the order of the process, in our case two extra periods.[1] Intuitively (but not necessarily), the coefficients on lags of shocks are lower than 1 so that the effect of the shock decreases over time. Of course, the length of the shock effect as well as its strength in each period can be easily changed with the order of the model or the size of the coefficients: moving average models give us complete flexibility over this, albeit at a cost of possible overparametrization.

An alternative to lengthening the effect of shock through increasing the order of moving average model is to use autoregressive model. AR models eliminate effects of shocks gradually, multiplicatively.[2] Therefore, in contrast to MA models, the effects of shocks persist for more periods than is the order model. In case of our lockdowns we can think of lasting lockdowns for most dangerous industries, say restaurants focused on international tourists (or similarly international air travel). Picture below captures such situation:

Of course we could combine moving average and autoregressive models to get ARMA model, which combines the profile of responses in first and second pictures:

Note that the different three options of lockdown effect profiles translate into slight, but important variation in growth rates we should observe. In first case large drop is followed by immediate and rapid growth for 3 months. In second case the growth is still immediate, but it is not rapid; instead the recovery is spread over many months. Finally, in the last case the growth is not even immediate but actually comes with some delay.

The previous pictures all assumed that there is no confidence shock associated with the pandemic. This was our collective assumption in January and February, but in March it became clear that we will not return to normal as soon as the lockdowns are over. Simply, the pandemic also caused global recession, which influences are current behavior through expectations about future economic situation, by for example lowering our expectations of future incomes (or even current incomes for the unlucky who lost their job already). This of course causes people to cut back on expenditures already now, and hence spending on restaurants would now be lower even if we would eliminate Covid-19 with a magic wand. To capture this, the next picture adds the confidence channel to the previous picture.[3] The key difference is that after lockdowns are over we do not return to the normal level of spending, but rather remain slightly below for the rest of the year. Correspondingly, the growth rates in May-July period are lower, but mild growth continues for the rest of the year.

Turning to econometric calibration, what we have added is AR(2) process for confidence. As before, we can see that the autoregressive model makes the effect last very long: the confidence component is negative throughout the whole year. Moreover, AR models of order higher than 1 with first coefficient bigger 1 also feature amplification: the peak effect occurs with delay. Therefore, the confidence component is largest in absolute value in June, two months after the initial shock. Finally, the sum of the coefficients is very close to 1, so that the effect of the shock is eliminated very slowly and even at the end of the year it is close to its maximum size.

Of course, things could be easily re-parametrized. We could ensure that the peak effect is either sooner or later, and we could ensure that the elimination is faster or slower. This could be done either within the context of AR(2) model, so would not require additional parameters, or by changing the order of the model. Examples of two re-parametrizations are below:

Therefore, the difference between autoregressive and moving average processes is in terms of trade-off between flexibility and over-parametrization: autoregressive process can give us some flexibility for given number of parameters, while moving average processes can give us infinite amount of flexibility at cost of many parameters.

Finally, all the charts above assumed that ¾ of the impact we observe in May is due to lockdowns and rest due to demand effects. Of course, for different consumption series different distribution will make sense: for other industries it will be mostly demand shock and bit of lockdown shock (spending on newspapers); for some it will be more equal; and for some there might be even a negative lockdown shock (think purchases of food in grocery stores).

Example 2 – Spending on cars: Lockdown shock with pent-up demand but no substitution

The second example applies to series which is affected by the lockdowns, but after lockdowns are lifted it benefits from the pent-up demand (but not from substitution effects).  An example could be spending in cars – car dealerships in many countries were closed for most of April, so that the spending on cars is substantially lower in that. At the same time, most people who were planning to buy car will do so soon after lockdowns anyway irrespective of the pandemic. Such situation is captured in next chart:

Here the crucial thing is to realize, that the pent-up demand gives us temporary small boost in the May (and even smaller in June), so that the rebound is faster than before. After June the profile is again determined solely by the confidence component and hence is identical as before. There is one even more important realization: since the pent-up demand gives us temporary boost in terms of level, there is actually decline in the spending in June, even though the June deviation from normal is smaller than before. This highlights the problematic nature of growth rates in coming months.

In terms of econometric calibration not much has changed: lockdown and confidence components are as in original case, while pent-up demand component is moving average model as the lockdown. The way things have been specified is using the same error as in lockdown equation and set current-period coefficient to zero, which is rather unusual. Nevertheless, this is for simplicity: one could write this down more elegantly, but it would blunt the message the pen-up demand is just partial reversal of lockdown shocks.

Example 3 – Spending on home-improvement: Lockdown shock with pent-up demand and substitution

The last example applies to series which is affected by the lockdowns, but after lockdowns are lifted it benefits from the pent-up demand as well as substitution effects.  An example could be spending on home improvement – this was impossible during the lockdown since hobby markets were closed for most of April, so that the spending on home improvement is substantially lower in that month. At the same time, most people who were planning to do such home improvement will do so soon after lockdowns anyway irrespective of the pandemic. And on top of that, since people saved money on some regular expenditures with which they do not plan to catch up – think no tourism, restaurant visits or cinema outings – they decide to use left-over money for home improvements. Such situation is captured in next chart:

 The picture resembles the previous case but the rebound is even stronger in May, so that the deviation from normal actually turns positive in this month. However, once the temporary positive effects dissipate the confidence effect takes over and the series goes back below normal. The growth rate in this situation is even more strangely looking than in previous case – after drop there is even larger jump, followed by large second drop before stabilization. Hence, growth rates do not tell useful story almost at all.

Econometrically, we have added another moving average process very similar to pent-up demand, so there is nothing new in that sense.

Conclusion – More caution than usual needed

Given the nature of the shock the behavior of individual consumption series will vary greatly. This divergence reflects the fact that in current situation – in contrast to normal recession – series can be thought of as composed of several differently-behaving components. This then can play havoc in how individual series will behave: not only will there be large swings in growth rates, there might be also multiple switches from positive to negative growth rate. This highlights how growth rates can be very misleading in coming months. While level will certainly be more informative, one can imagine reversals even here. So the overall message is that one has to be cautious when drawing conclusions from  individual consumption series in coming months.

---

[1] Of course, one could argue that what is really happening is three successive shocks, but this distinction is relatively slight in current situation. The notion here is that the effects of lockdowns always linger for some time, so that they can be described by some firm process.

[2] In the simplest case of AR(1) model, say with coefficient 0.9, 10% of the previous period effect is eliminated in each period.

[3] Note that the April impact is the same as before, just that part of it is demand shock. This is true throughout this document.