2.1. Decomposing Covid-19 data into a trend, seasonal and irregular component

Define $ {I}_t $ as the cumulative number of positive tests and $ {i}_t $ for the daily number of positive tests, where $ {i}_t=\Delta {I}_t $ , and equivalently, the cumulative number of deaths is $ {D}_t $ , and the daily count is $ {d}_t=\Delta {D}_t $ , where we let $ {Y}_t $ denote $ {I}_t $ or $ {D}_t $ . Our forecasting models are for the logarithm of $ {Y}_t $ , adding 1 to allow for a zero count at the beginning of the pandemic. The resulting decomposition into trend $ {\hat{\mu}}_t $ , seasonal $ {\hat{\gamma}}_t $ and remainder $ {\hat{\varepsilon}}_t $ is

$$ \log \left({Y}_t+1\right)=\log {\hat{\mu}}_t+\log {\hat{\gamma}}_t+{\hat{\varepsilon}}_t, $$

which is transformed back using $ {\hat{u}}_t=\exp \left({\hat{\varepsilon}}_t\right) $ :

$$ {Y}_t={\hat{\mu}}_t{\hat{\gamma}}_t{\hat{u}}_t-1=\left[{\hat{\mu}}_t-1\right]{\hat{\gamma}}_t{\hat{u}}_t+\left[{\hat{\gamma}}_t{\hat{u}}_t-1\right]\approx \left[{\hat{\mu}}_t-1\right]{\hat{\gamma}}_t{\hat{\mu}}_t={\hat{Y}}_t{\hat{\gamma}}_t{\hat{u}}_t. $$

Both $ {\hat{\gamma}}_t $ and $ {\hat{u}}_t $ have an expectation of one and are uncorrelated.

The decomposition is obtained by taking moving windows of the data and saturating these by segments of linear trends, denoted trend indicator saturation (TIS; Castle et al., Reference Castle, Doornik, Hendry and Pretis2019). The changing seasonality is modelled by including six indicator variables for 6 days of the week and both a weekly sine and cosine wave, and a half-weekly sine and cosine wave. Despite redundancy, these can all be included initially as selection is applied. Sparsity is obtained by selecting the broken trends and seasonal components that are significant at tight significance levels using the machine learning tree-search algorithm, Autometrics (Doornik, Reference Doornik, Castle and Shephard2009). An estimate of the unobserved flexible trend and unobserved changing seasonal pattern is obtained by taking the average of the fitted values for each observation across all windows that include that observation. The remainder is the difference between the actual observation and the estimated trend and seasonal component.

2.2. The Cardt forecasting device

The trend and remainder terms are forecast separately using Cardt, and recombined, adding in the seasonal component from the last observations at the seasonal frequency (e.g., the seasonal pattern from the last week of in-sample data for the daily Covid-19 data is extrapolated forwards) in a final forecast. Cardt is applied to the irregular component, as there may be some residual dynamics that are captured in the decomposition. The remainder is not a martingale-difference process.

To apply Cardt, three models are estimated including:

$ \delta $ : Obtains estimates of the growth rate based on first differences, but is dampened by removing large values and allowing for seasonality.

$ \rho $ : Estimates a simple autoregressive model with seasonality, forcing a unit root if the estimates are close to one and hence switching to a model in first differences with dampened mean.

THIMA: A trend-halved integrated moving average model, which consists of a dampened trend which is arbitrarily halved, together with an intercept correction estimated by a moving average model.

The arithmetic mean average of the three forecasts is computed. These forecasts are then calibrated by treating them as if they were observed, and a richer autoregressive model is estimated from the extended data series. The fitted values from this calibrated model give the final forecasts, undoing any transformations such as log and differencing. Higher orders of integration [e.g., I(2) and damped I(2)] can be allowed for when applying the methodology; Doornik et al. (Reference Doornik, Castle and Hendry2020b) provides further details.

The results from the M3 and M4 competition data, which include data of differing sample sizes, frequencies, category of data and degree of nonstationarity (both stochastic trends and abrupt shifts), suggest that the Cardt method forecasts well over short horizons. The method dampens trends and growth rates, which is important to avoid wild forecasts, averages across forecasts (which is a principle dating back to Bates and Granger, Reference Bates and Granger1969) and robustifies the forecasts to breaks in the data by ‘over-differencing’ (see Hendry, Reference Hendry2006). We next apply the forecasting method to Covid-19 cases and deaths in Section 3 and UK unemployment in Section 4.

3.1. Evaluation of statistical forecasts in comparison to epidemiological forecasts

We compare the evolution of the forecast performance of our statistical methods with those from the Los Alamos National Laboratory (LANL), which have been published twice a week since 5 April 2020, and forecasts from the Institute for Health Metrics and Evaluation (IHME), published since 25 March 2020 but not consistently.Footnote ³ The LANL forecasting model is not a full susceptible, infected and recovered epidemiological model, but is a modification of one in which they use a statistical model to capture the dynamics of the infection rate and then they map this to the reported data. They also produce uncertainty bands capturing model and measurement uncertainty.

Figure 5 records the forecast paths for the UK comparing our forecasts with those from LANL, with cases on the left panel and deaths on the right panel, up to November 2020. There are substantial revisions to both cases and deaths data. For confirmed cases, the UK data include results from both pillar 1 and pillar 2 testing. Pillar 1 testing includes those with a clinical need, and health and care workers, whereas pillar 2 testing is for the wider population. Up to 1 July, these data were collected separately meaning that people who had tested positive via both methods were counted twice. On 2 July, data for both pillars were combined, and around 30,000 duplicates were found and removed from the data, hence the revision to reported cases on 2 July. For the data on deaths, there was a large revision on 12 August. Deaths were redefined, so that a Covid-19-related death was recorded only if the individual had a positive test within the last 28 days. Previously, all deaths after a positive test were attributed to Covid-19. The data were then revised backwards. Time-series models have the advantage of adjusting to such data revisions rapidly, without the need for intercept adjustments as structural models would.

Figure 5. (Colour online) Forecast paths for UK, cases (left panel) and deaths (right panel), with our Cardt forecasts in red and LANL forecasts in blue

Source: www.doornik.com/COVID-19.

Table 1 evaluates the two forecast paths, reporting the mean absolute percentage error (MAPE) and root-mean-square percentage error (RMSPE), as well as the percentage of forecasts lying below the 10 per cent and above the 90 per cent quintiles, where $ F $ denotes our forecasts. Let $ {\hat{y}}_{j,T+h} $ denote a forecast at horizon $ h $ (where $ h $ spans $ 1-7 $  days ahead) from group $ j=1,\dots, J $ :

$$ {\displaystyle \begin{array}{l}\hskip0.24em \mathrm{MAPE}=\frac{100}{J}\sum \limits_{j=1}^J\frac{\left|{y}_{j,T+h}-{\hat{y}}_{j,T+h}\right|}{y_{j,T+h}},\\ {}\mathrm{RMSPE}={\left[\frac{100}{J}\sum \limits_{j=1}^J{\left({y}_{j,T+h}-{\hat{y}}_{j,T+h}\right)}^2\right]}^{1/2},\end{array}} $$

where $ {y}_{j,T+h}>0 $ in our application. For confirmed cases, both Cardt and LANL forecasts are close, although Cardt does slightly better on RMSPE, but for deaths, Cardt tends to outperform those of LANL by a margin. The interval forecasts are much harder to accurately predict for both sets of forecasts. For the bottom quantile, our forecasts are significantly below 10 per cent but a bit closer for the top quantile, whereas the LANL forecasts are closer but with considerable variation.

Table 1. Forecast accuracy for $ {I}_t $ and $ {D}_t $ for UK data spanning 30 April 2020 to 28 October 2020. There are 51 forecast errors at each horizon

Abbreviations: LANL, Los Alamos National Laboratory; MAPE, mean absolute percentage error; RMSPE, root-mean-square percentage error.

We next compare our forecasts to those from IHME, which use forecasting models that are a hybrid of disease transmission models and statistical models. Since the change in the definition of a Covid-19-related death, the IHME have been targeting a different measure of deaths to that reported by Johns Hopkins, but we evaluate the forecasts based on their definition of the outturns, which are reported the following week. Table 2 records equivalent statistics for a comparison between our forecasts $ F $ and the IHME forecasts, although over a smaller sample when the forecast dates coincide. The Cardt forecast errors are consistently smaller over 1–7 days ahead, but again the quantiles are hard to predict.

Table 2. Forecast accuracy for $ {D}_t $ for UK data spanning 10 April 2020 to 2 August 2020. There are 18 forecast errors at each horizon

Abbreviations: IHME, Institute for Health Metrics and Evaluation; MAPE, mean absolute percentage error; RMSPE, root-mean-square percentage error.

Thus, our statistical forecasts for Covid-19 perform well and can rapidly update when there are changes in data definitions or shifts in the data. We next examine how the method performs when forecasting UK unemployment during the pandemic.

4.1. A model of aggregate UK unemployment

We next derive a theoretically justified, data-based, econometric model for unemployment to compare its forecast performance to that of Cardt. $ {U}_t $ is the outcome of supply and demand for labour, aggregated across all prospective workers, with labour demand derived from demand for goods and services. This implies a highly complex DGP, so instead we use a profits proxy, denoted $ \pi $ , which assumes that unemployment falls when hiring labour is profitable, and increases if it is not profitable. $ \pi $ measures the gap between the real interest rate (reflecting the costs) and the real growth rate (reflecting the demand side), such that the unemployment rate rises when the real interest rate exceeds the real growth rate, and vice versa:

(1) $$ {\pi}_t=-\left[{R}_{l,t}-{\Delta}_{12}{p}_t-{\Delta}_{12}{y}_t\right], $$

where $ {R}_{l,t} $ is the long-term interest rate; $ {\Delta}_{12}{y}_t $ is the annual change in log Gross Value Added which measures GDP at the monthly frequency and $ {\Delta}_{12}{p}_t $ is the annual Consumer Price Index inflation rate, all recorded in figure 7 for the in-sample period up to 2019(12).Footnote ⁴

Figure 7. Panel (a): annual change in log GDP. Panel (b): annual CPI inflation rate. Panel (c): long-term (10 year) government bond yields. Panel (d): profits proxy measured by $ {\pi}_t=-\left[{R}_{l,t}-{\Delta}_{12}{p}_t-{\Delta}_{12}{y}_t\right] $ , for 1997(4)–2019(12)

Other regressors in the model are recorded in figure 8 and include annual nominal wage inflation ( $ {\Delta}_{12}w $ ) in panel (a) along with real wage inflation which was negative for almost 8 years after the financial crisis. Panel (b) records the log of average real weekly earnings and output per worker, with the resulting wage share given in panel (c), adjusted to give a zero mean by calculating the in-sample mean of the wage share, denoted $ \hat{\mu} $ . Panel (d) records the output gap ( $ {y}^{gap} $ ), measured using the deviation between the log of output and the fitted value from impulse indicator saturation (IIS; Hendry et al., Reference Hendry, Johansen and Santos2008) and TIS (Castle et al., Reference Castle, Doornik, Hendry and Pretis2019), estimated to the end of 2019. TIS attributes the fall in output over the financial crisis to mostly shifts in permanent or potential output, so the output gap over this period is fairly small.

Figure 8. (Colour online) Panel (a): nominal and real wage inflation. Panel (b): output per worker and real wages. Panel (c): the wage share, measured by $ w-p-y+l-\hat{\mu} $ . Panel (d): output gap, measured as deviation from fitted regression of IIS and TIS over sample to 2019(12) selected at $ \alpha =0.0001 $ , for 2000(1)–2019(12)

We specify an autoregressive distributed lag model which initially includes $ {\pi}_{t-i} $ , $ {\Delta}_{12}{w}_{t-i} $ , $ {y}_{t-i}^{gap} $ and $ {\left(w-p-y+l-\hat{\mu}\right)}_{t-i} $ for $ i=0,\dots, 13 $ , seasonal dummies, IIS and step indicator saturation (see Castle et al., Reference Castle, Doornik, Hendry and Pretis2015), and nonlinear transformations of regressors given by $ {\left({x}_j-{\overline{x}}_j\right)}^k-\overline{{\left({x}_j-{\overline{x}}_j\right)}^k} $ for $ k=2,3 $ , and $ {x}_j\in \left\{{\pi}_{t-i};{\Delta}_{12}{w}_{t-i};{y}_{t-i}^{gap};{\left(w-p-y+l-\hat{\mu}\right)}_{t-i}\right\} $ , where $ \overline{\cdot} $ indicates the sample mean. The nonlinear transformations are polynomial in form, as many nonlinear models, including regime-switching and smooth-transition regression models, which are popular in the unemployment literature, can be approximated by Taylor expansions, and so polynomials form a flexible approximating class, but must enter in deviations from means (by demeaning both prior to and after the polynomial transformation) to avoid high levels of collinearity (see Castle and Hendry, Reference Castle, Hendry, Wang, Garnier and Jackman2011). An encompassing test could be used against specific nonlinear models like threshold specifications as in Castle and Hendry (Reference Castle, Hendry, Haldrup, Meitz and Saikkonen2014).

This results in 621 candidates with 215 observations. We retain the primary economic regressors, constant and seasonals (81 parameters), and select the saturation estimators and nonlinearities at a significance level of $ \alpha =0.0001 $ using Autometrics (Doornik, Reference Doornik, Castle and Shephard2009). We then select over the regressors at $ \alpha =0.001 $ . The resulting selected model isFootnote ⁵

(2) $$ {\displaystyle \begin{array}{l}{\hat{U}}_t=\underset{(0.0005)}{0.0008}+\underset{(0.006)}{0.983}{U}_{t-1}-\underset{(0.007)}{0.028}{\pi}_t+\underset{(0.008)}{0.036}{\pi}_{t-2}-\underset{(0.007)}{0.022}{\Delta}_{12}{w}_{t-2}\\ {}\hskip2em +\underset{(0.004)}{0.025}{\left(w-p-y+l-\hat{\mu}\right)}_{t-2}+\mathrm{seasonals}\\ {}\hskip2em \hat{\sigma}=0.09\%;\hskip0.5em {\mathtt{F}}_{ar}\left(\mathrm{7,191}\right)=1.00;\hskip0.5em {\mathtt{F}}_{ar ch}\left(\mathrm{7,201}\right)=0.55;\hskip0.5em {\chi}^2(2)=0.87;\hskip0.5em \\ {}\hskip2em {\mathtt{F}}_{hetero}\left(\mathrm{21,193}\right)=1.12;\hskip0.5em {\mathtt{F}}_{reset}\left(\mathrm{2,196}\right)=1.18;\hskip0.5em T=2002(2)-2019(12)\end{array}} $$

with the solved long-run solution

(3) $$ \hat{d}=U-\underset{(0.016)}{0.049}-\underset{(0.27)}{0.46}\pi +\underset{(0.36)}{1.32}{\Delta}_{12}w-\underset{(0.47)}{1.49}\left(w-p-y+l-\hat{\mu}\right). $$

The selected model is well specified, passing all diagnostic tests, and fits the in-sample data well, recorded in figure 9 with scaled residuals, residual density and residual autocorrelation function. The model includes the lagged unemployment rate picking up inertia, the profits proxy which enters contemporaneously and lagged two periods, nominal wage inflation and the wage share, both lagged two periods. No impulse or step indicators are retained, so there are no outliers or shifts in the data that are not explained by regressors in the model. This is quite remarkable given the sample includes the Financial Crisis and Great Recession period, where the regressors in the model are able to explain the impact on unemployment well. We also do not retain any nonlinear terms, refuting some empirical studies that argue that the unemployment rate is best characterised by regime switching behaviour. The solved out long-run solution gives an equilibrium unemployment rate of about 5 per cent. This matches the mean unemployment rate over the last 160 years. The results are similar to the model of unemployment reported in Hendry (Reference Hendry2001), which is estimated on nonoverlapping data of a different frequency.

Figure 9. (Colour online) Panel (a): unemployment rate and model fit. Panel (b): scaled residuals. Panel (c): residual density. Panel (d): residual autocorrelation

The coefficient on the lagged unemployment rate suggests $ U $ is close to a unit root. With an infinite sample, the unemployment rate is bounded between 0 and 1 and therefore could not contain a stochastic trend, but we have a fairly small sample of monthly data which does suggest local nonstationarity. We therefore transform to a stationary representation using (3), recorded in figure 10 [panel (d)]. The resulting new general model for $ \Delta {U}_t $ includes 12 lags of the differenced regressors and is reselected at a significance level of $ \alpha =0.001 $ using Autometrics. The intercept and seasonals are not selected over. Nonlinear functions and impulse and step indicators are not included given their absence in (2). The final model is

(4) $$ {\displaystyle \begin{array}{l}{\hat{\Delta U}}_t=+\underset{(0.0002)}{0.0003}+\underset{(0.067)}{0.20}\Delta {U}_{t-1}-\underset{(0.009)}{0.024}\Delta {\pi}_t-\underset{(0.002)}{0.013}{\hat{d}}_{t-1}+\mathrm{seasonals};\\ {}\hskip3em \hat{\sigma}=0.09\%;\hskip0.5em {R}^2=0.65;\hskip0.5em {\mathtt{F}}_{ar}\left(\mathrm{7,193}\right)=1.54;\hskip0.5em {\mathtt{F}}_{ar ch}\left(\mathrm{7,201}\right)=1.06;\hskip0.5em \\ {}\hskip3em {\chi}^2(2)=0.85;\hskip0.5em {\mathtt{F}}_{hetero}\left(\mathrm{17,197}\right)=0.94;\hskip0.5em {\mathtt{F}}_{reset}\left(\mathrm{2,198}\right)=1.93.\hskip0.5em \end{array}} $$

Figure 10. (Colour online) Panel (a): annual change in the unemployment rate and model fit. Panel (b): scaled residuals. Panel (c): residual density. Panel (d): $ \hat{d} $ from (3)

The model fit, scaled residuals and residual density are recorded in figure 10, along with $ \hat{d} $ from the long-run solution reported in (3) in panel (d). The model is congruent in-sample, with short-run dynamics due to changes in the profits proxy and past changes in the unemployment rate, with a speed of adjustment back to equilibrium of 1.3 per cent per month.

4.2. Forecasting the UK unemployment rate over the pandemic

We commence forecasting in January 2020 before the pandemic took hold in the UK,Footnote ⁶ up to the last available unemployment rate observation for the 3-month average over August–October 2020, recorded as the rate for September, resulting in nine forecast observations. We produce conditional forecasts from 1- to 3-month ahead from the econometric model (2), conditioning on contemporaneous data ( $ {\pi}_t $ ), where the 1-month ahead forecasts will include a measure of the unemployment rate for that month in the data used to forecast, given the 3-month averaging, so are interpreted as partial nowcasts. The model parameters are fixed at their in-sample estimates, so there is no recursive updating through the forecast period.

Figure 11 [panel (a)] records the forecasts with 95 per cent error bands, which show the forecasts perform poorly over the first wave of the pandemic. The forecast commencing in February predicts a strong uptick in unemployment in March and April due to the decline in output reflecting a weakening demand side in the profits proxy. The model predicts unemployment to be rising significantly throughout March, April and May, when unemployment remained low. By June, the outturns are closer to the 1-step ahead forecasts made in May, although the forecasts then start to underpredict the rise in unemployment at the 2- and 3-month horizons. The last 1-step ahead forecast for September 2020 made in August 2020, marked by the solid black square, includes data on the unemployment rate in September. The accuracy of this forecast demonstrates the benefits of conditioning on current information, although the poor 1-step ahead forecasts earlier in the sample suggests that this is not always the case. The UK’s first nationwide lockdown extended from 23 March to 10 May, coinciding with the forecast failure from the econometric model.

Figure 11. (Colour online) Panel (a): conditional 1–3-month ahead forecasts from the econometric model. Panel (b): conditional 1–3-month ahead forecasts from the econometric model with a lockdown dummy. Panel (c): unconditional 1–3-month ahead forecasts using Cardt. Panel (d): equally weighted average of the econometric model forecasts and Cardt forecasts

4.2.1. Understanding the forecast failure

Figure 12 records the extended data series up to 2020(9), where the macroeconomic impact of the pandemic is huge. Panel (a) shows annual falls of 28 per cent in April and 26 per cent in May in gross value added. Such falls absolutely dominate the historical scale of growth rates, and are reflected in the profits proxy in panel (c), as well as the wage share [panel (e)] and the output gap [panel (f)].

Figure 12. (Colour online) Panel (a): annual change in log gross value added and annual CPI inflation rate. Panel (b): long-term (10-year) government bond yields. Panel (c): profits proxy measured by $ {\pi}_t=-\left[{R}_{l,t}-{\Delta}_{12}{p}_t-{\Delta}_{12}{y}_t\right] $ . Panel (d): nominal and real wage inflation. Panel (e): the wage share, measured by $ w-p-y+l-\hat{\mu} $ . Panel (f): the output gap computed by extrapolating the trend estimated to 2019(12) and calculating the deviation from actual output over 2020. Sample: 2019(1)–2020(9)

Extending the in-sample period to September 2020 and re-estimating (2) results in the coefficient on $ {\pi}_t $ falling to $ {\hat{\beta}}_{\pi }=-0.003 $ with $ \mid {\hat{t}}_{\pi}\mid =1.09 $ , so becomes insignificant. Table 3 records the correlation between $ {\Delta}_{12}U $ and $ \pi $ for the in-sample and forecast periods. The correlation is effectively zero over 2020. To disentangle this effect, we apply a subset of multiplicative indicator saturation (MIS; see Castle et al., Reference Castle, Hendry and Martinez2017) to identify parameter nonconstancy in the model. We include $ {\pi}_t\times {S}_{2020(j)} $ for $ j=1,\dots, 9 $ , where $ {S}_{2020(j)} $ is a step indicator that takes the value 1 for observations 2020(1)–2020( $ j $ ) [2020(9) is the end of the sample], in model (2). All regressors in the model are fixed apart from the interaction terms, and we select at $ \alpha =0.001 $ using Autometrics. $ {\pi}_t\times {S}_{2020(3)} $ is retained leaving the full sample $ {\pi}_t $ coefficient close to that from (2) using data to 2019(12). The method successfully detects the induced shift in our estimated model following the policy intervention which began in March, and reveals that the apparent change in the coefficient of $ \pi $ is due to the shift in policy.

Table 3. Correlation between $ \pi $ and $ {\Delta}_{12}U $

Forecast failure is often due to structural breaks in the data that are not captured by the economic model. Here, the model predicts a structural break in the data which does not materialise. This is because policy intervention changes the earlier constant relationships captured in the economic model by artificially holding down measured unemployment rates via the furlough scheme and other economic policies. We view these conditional forecasts from the economic model as scenario forecasts answering the question ‘what would the unemployment rate have been if the policy intervention had not occurred?’

4.2.2. Forecasting with a policy intervention dummy

MIS revealed a significant indicator in March which we use to capture the effect of the UK’s Covid-19 policies. Over the forecast period, the UK underwent its first national lockdown, with accompanying economic policies. Lockdown restrictions requiring nonessential businesses to close were imposed on 23 March 2020, and the UK announced a job retention scheme, aimed to support employers who could not maintain the current workforce, because their operations were affected by coronavirus. The scheme, also known as the furlough scheme, paid 80 per cent of workers’ salaries. The aim of the furlough policy was to mitigate the impact of lockdown on recorded unemployment, so will have a direct effect on our forecasting model. We take the econometric model (2), fixing the in-sample parameters, and add a ‘lockdown’ dummy given by

$$ {\displaystyle \begin{array}{l}{D}_f=1\hskip0.72em \mathrm{for}\ \mathrm{March},\mathrm{April},\mathrm{May};\\ {}{D}_f=0\hskip0.82em \mathrm{otherwise}.\end{array}} $$

Alternative taperings could be considered, as the economic impact will be due to changing behaviour as well as government policy. However, behavioural changes are harder to measure, and so linking dummies to explicit policies holds appeal. This also means that the model can be tested over the second and third national UK lockdowns when the unemployment data become available.

Although lockdown was only introduced on 23 March, the evidence from MIS suggests that there is a sufficient shift in March to commence forecasting in April using data up to March. Estimates of the lockdown dummy are given in table 4. The dummy is poorly estimated on March data alone, so there is not enough information to improve the forecast for April. However, estimating the model up to April leads to a highly significant lockdown dummy which heavily adjusts the forecasts of unemployment. By May, there are stable estimates of the policy intervention dummy, dampening the forecasts from the model without policy intervention.

Table 4. Parameter estimates for the lockdown dummy included in (2)

The forecasts for 1–3-month ahead are recorded in figure 11 [panel (b)]. The first three sets of forecasts are identical to those in panel (a). The next set of forecasts made in March show a small reduction in forecast error relative to the unadjusted model (an average forecast error over $ h=\mathrm{1,2,3} $ of 1 per cent compared to a forecast error of 1.26 per cent for the unadjusted model). With just one more observation to estimate the lockdown dummy, the improvement in forecast accuracy is substantial (averaging over the three forecast horizons, the forecasts made in April have an average forecast error of 0.18 per cent relative to 0.67 per cent for the unadjusted model). The forecasts are the same after the lockdown dummy ends, but during lockdown, the adjustment significantly improves the forecast performance of the econometric model.

To get a sense of the magnitude of the forecast differences, in the 3-month period from April to June 2020, there were 1.338-million people unemployed in the UK. In April, the econometric model predicted there would be 1.546-million people unemployed for the same period.Footnote ⁷ The same model including the lockdown dummy predicted 1.377-million unemployed, a difference of 169,000 people. Interpreting this difference as a scenario, whereby the level of unemployment would have been 169,000 higher had furlough and other related policies not been implemented, suggests that the economic response was fundamental to holding unemployment down in the first wave of the pandemic.

4.3. Unconditional econometric forecasts

The forecasting model (2) conditions on contemporaneous data, $ {\pi}_t $ , and hence a comparison with unconditional forecasts will be based on different information sets. To make the econometric model forecasts unconditional, we replace the known $ {\pi}_{T+h} $ , where $ T $ is the forecast origin and $ h $ is the forecast horizon, with forecasts of $ {\hat{\pi}}_{T+h\mid T} $ using Cardt to forecast the profits proxy. Castle et al. (Reference Castle, Doornik and Hendry2018) examine when contemporaneous regressors should be retained in forecasting models if they also need to be forecast, given that structural breaks occur in the conditioning variables. Despite conditioning on a subset of the information set available for the conditional econometric model forecasts, table 5 shows that the unconditional forecasts are more accurate over March and April when the pandemic took hold. The extrapolative Cardt forecasts of $ {\pi}_t $ were poor in March and April when profits fell dramatically. $ {\hat{\pi}}_t $ missed the fall initially, but these poor forecasts helped the econometric model to avoid predicting a large rise in the unemployment rate. The forecast error in the profits proxy is a measure of the economic policies implemented in March and April. However, moving forward to May and June, forecasts of the profits proxy miss the rebound, predicting very large negative values. This impacts on the econometric model forecasts, leading the unconditional forecasts to perform much worse. The comparison of conditional and unconditional forecasts show that more information does not always lead to improved forecast performance as structural breaks impact on the forecasts in different ways.

Table 5. Absolute forecast errors ( $ \times 100 $ ) for unemployment forecasts over 2020. Unconditional econometric are unconditional forecasts from the econometric model using Cardt to forecast the contemporaneous profits proxy $ {\pi}_t $ . Average is the equally weighted average of the conditional econometric and Cardt forecasts. Bold indicates smallest absolute forecast errors

Abbreviations: MAPE, mean absolute percentage error; MPE, mean percentage error.

4.4. Forecasting the unemployment rate using Cardt

The econometric model forecasts can be used as a counterfactual to measure what effect the pandemic would have had on unemployment if the economic policies that accompanied the first lockdown had not been implemented. We can think of the econometric model as describing ‘business as usual’ as the model specification and estimation is fixed prior to the pandemic, thus reflecting the predicted unemployment rate had no mitigation policies, including furlough, been implemented. To contrast, we use the extrapolative statistical forecasting device, Cardt, to forecast unemployment. This device does not account for any specific economic and policy measures implemented over the forecast horizon but relies on extrapolating recent trends in the data. The method worked well for the Covid-19 data, so it is of interest whether it also produces reasonable forecasts for the unemployment rate. The forecasts are recorded in figure 11 [panel (c)]. The statistical forecasts are much closer to the outturns until June when the unemployment rate starts to rise, but the extrapolative forecasts do not. In April, Cardt predicted 1.321-million unemployment in the three months from April to June 2020, which is a difference of 225,000 people compared to the ‘business as usual’ econometric model.

Cardt also provides a useful benchmark against published forecasts. The Bank of England (BoE) unemployment rate forecasts from the Monetary Policy Report in January 2020Footnote ⁸ predict an unemployment rate of 3.8 per cent for 2020Q1, very close to our Cardt forecasts. In the May Monetary Policy Report,Footnote ⁹ the BoE offer a scenario with the unemployment rate rising to almost 10 per cent before falling back in 2021, and by August,Footnote ¹⁰ they predict a rate of 7.5 per cent in 2020Q4. Cardt predicts an unemployment rate of 4.7 per cent for 2020Q4 based on data available to August 2020, substantially lower than the forecast produced by the BoE, and a difference of 960,000 people (assuming a constant economically active population at September 2020 levels). Being data-based, Cardt forecasts are influenced by the furlough having reduced measured unemployment.

4.5. Forecast comparisons

Table 5 records the absolute forecast errors for the conditional econometric model with contemporaneous $ {\pi}_t $ , the unconditional econometric model with forecast $ {\hat{\pi}}_t $ , the conditional model with the lockdown policy dummy and the statistical forecasts. The row ‘Average’ reports the equally weighted forecasts from the conditional econometric model and Cardt, with the forecasts recorded in figure 11 [panel (d)]. MAPE denotes the mean absolute percentage error for a given horizon across all months in the forecast period, and MPE denotes the mean percentage error.

The statistical model produces some of the most accurate forecasts up to and including May for all horizons, with substantial reductions in forecast errors relative to the conditional econometric model throughout the lockdown period. The econometric model forecasts are poor in April and May when policies such as furlough played a significant role, and could be seen as the counterfactual unemployment rate had such a policy not been introduced. Accounting for the policy takes some time, but it can improve the econometric model forecasts, as seen for the two months when the policy dummy can be accurately estimated. For 3-month ahead, the lockdown policy dummy has no effect throughout April and May because of the 3-month lead time, but it does improve the forecasts for June and July. The average is never the best forecast (except for the 3-month ahead forecast in July), but both minimizes the risk of very large forecast errors and has the smallest MPE. The unconditional econometric forecasts are the best more than 40 per cent of the time, but also occasionally the worst.

Ericsson (Reference Ericsson1992) proposes a forecast encompassing test that assesses the ability of one set of forecasts to explain the errors of another forecasting device allowing for the forecasts to be cointegrated. This builds on the encompassing principle developed by Mizon and Richard (Reference Mizon and Richard1986) and the test of forecast encompassing by Chong and Hendry (Reference Chong and Hendry1986; see also Hendry, Reference Hendry1988). Let $ {\hat{y}}_{t\mid t-j} $ and $ {\tilde{y}}_{t\mid t-j} $ denote two sets of forecasts from the models in table 5, with $ {\hat{e}}_t={y}_t-{\hat{y}}_{t\mid t-j} $ and $ {\tilde{e}}_t={y}_t-{\tilde{y}}_{t\mid t-j} $ as the forecast errors, where $ j=\mathrm{1,2,3} $ , and $ t=2020(1)-2020(9) $ . The forecast encompassing tests for both directions are given by

(5) $$ {\displaystyle \begin{array}{l}{\hat{e}}_t={\lambda}_0+{\lambda}_1\left({\tilde{y}}_{t\mid t-j}-{\hat{y}}_{t\mid t-j}\right)+{\eta}_{1,t},\\ {}{\tilde{e}}_t={\delta}_0+{\delta}_1\left({\hat{y}}_{t\mid t-j}-{\tilde{y}}_{t\mid t-j}\right)+{\eta}_{2,t}.\end{array}} $$

If $ {\mathrm{H}}_0:{\lambda}_1=0 $ is rejected, the difference between $ {\tilde{y}}_{t\mid t-j} $ and $ {\hat{y}}_{t\mid t-j} $ helps to explain the forecast errors from model $ \hat{\cdot} $ , and hence $ {\tilde{y}}_{t\mid t-j} $ provides information above and beyond what is available from the $ {\hat{y}}_{t\mid t-j} $ forecast. Likewise, if $ {\mathrm{H}}_0:{\delta}_1=0 $ is rejected, we conclude that $ {\hat{y}}_{t\mid t-j} $ forecast encompasses $ {\tilde{y}}_{t\mid t-j} $ .

The encompassing test is normally distributed as $ T,H\to \infty $ , but our sample size is very small. Hendry (Reference Hendry1986) provides power functions of the Chong and Hendry (Reference Chong and Hendry1986) encompassing test for $ H=5,\dots, 14 $ , for the case when both forecasting models are misspecified and finds a fair degree of power. Gaussianity of the disturbances in (5) is required for small sample analysis using the t-distribution, but that assumption does not hold for our forecasts. As a check, we conduct a simulation study to evaluate the power of the encompassing test, reported in table 6.Footnote ¹¹ For the forecast encompassing case where a second forecasting model provides no additional information over and above the forecasts from the first model, the power to detect forecast encompassing declines substantially for small samples. When neither model forecast encompasses the other as both forecasts yield additional information, the power is still quite high even in very small samples, and does not fall by much under non-normal errors. These results are close to the power estimates in table 3 of Hendry (Reference Hendry1986). Therefore, we proceed with the empirical encompassing tests, but note that the power may be low given the small sample.

Table 6. Probability of rejection of the null hypothesis for the Ericsson (Reference Ericsson1992) encompassing test

Table 7 records the results from the forecast encompassing tests. For each row, forecasts from model M1 are denoted $ {\hat{y}}_{t\mid t-j} $ in (5) and forecasts from model M2 are denoted $ {\tilde{y}}_{t\mid t-j} $ . In the first row, there is evidence that the unconditional model forecasts provide information above that contained in the conditional model forecasts. This is striking as the unconditional forecasts use a reduced information set with forecasts of the profits proxy. There is no strong evidence that the conditional model forecasts encompass the unconditional model forecasts, so they do not capture additional information relative to the unconditional forecasts. The following three rows test various specifications of the econometric model against the statistical forecasts. There is statistically significant evidence that Cardt forecast encompasses the econometric model at all forecast horizons. Hence, there is a clear benefit to using statistical forecasts during periods subject to structural breaks.

Table 7. Forecast encompassing tests statistics for 2020 unemployment rate forecasts. Coefficient estimates from (5) are reported with estimated standard errors in parentheses and estimated t( $ df $ )-statistics in square brackets. $ df=7 $ for 1 month, $ df=6 $ for 2 months and $ df=5 $ for 3 months. $ {}^{\ast } $ , $ {}^{\ast \ast } $ and $ {}^{\ast \ast \ast } $ denote significance at 10 per cent, 5 per cent and 1 per cent, respectively

As a benchmark comparison, we consider the same forecasts over 2019, recorded in figure 13 with table 8 reporting the forecast errors and table 9 reporting the forecast encompassing test statistics. All forecasts are more accurate given the stable environment, but the conditional econometric model forecasts produce the smallest forecast errors on average. The unconditional forecast errors are close to those of the conditional forecast errors, so the costs of forecasting contemporaneous regressors using Cardt are small, and negative in some cases where the unconditional forecasts deliver the smallest forecast errors of all models considered. At the 2- and 3-month horizons, there is evidence that both the conditional and unconditional forecasts encompass each other, so both yield additional information. This implies the Cardt forecasts of the profits proxy do contribute additional information. The forecast errors from Cardt are mostly larger than those for the econometric model, although table 9 shows that they also forecast encompass the econometric model. There is additional information in both the econometric model forecasts and the statistical forecasts, with significant forecast encompassing tests in both directions. As all forecast errors are in the same direction, averaging does not help.

Figure 13. (Colour online) Panel (a): conditional 1–3-month ahead forecasts from the econometric model over 2019. Panel (b): unconditional 1–3-month ahead forecasts using Cardt over 2019

Table 8. Absolute forecast errors ( $ \times 100 $ ) for unemployment forecasts over 2019. Unconditional are unconditional forecasts from the econometric model using Cardt to forecast the contemporaneous profits proxy $ {\pi}_t $ . Average is the equally weighted average of the conditional econometric and Cardt forecasts. Bold indicates smallest forecast errors

Abbreviations: MAPE, mean absolute percentage error; MPE, mean percentage error.

Table 9. Forecast encompassing tests statistics for 2019 unemployment rate forecasts. Coefficient estimates from (5) are reported with estimated standard errors in parentheses and estimated t( $ df $ )-statistics in square brackets. $ df=10 $ for 1 month, $ df=9 $ for 2 months and $ df=8 $ for 3 months. $ {}^{\ast } $ , $ {}^{\ast \ast } $ and $ {}^{\ast \ast \ast } $ denote significance at 10 per cent, 5 per cent and 1 per cent, respectively

The switch in rankings from 2019 to 2020, and the shift to unidirectional forecast encompassing for Cardt, highlights the value of different forecasting methodologies at different points in the forecast period due to the impact of structural breaks and policy interventions. The substantial and significant forecast improvements using Cardt over the lockdown period mean that over the forecast sample, this statistical forecasting method performs best overall, but in quiescent periods, a model embodying our theoretical understanding of the economy yields advantages. If the most appropriate forecasting model is not known, pooling using an equally weighted average can reduce large forecast errors if the forecasting models are differentially biased.