The visual inspection of the data given in the previous section highlighted that electricity consumption in 2020 exhibited very similar patterns to those in the previous 5 years, at least up to the day in which the first containment measures were introduced. Therefore, the information in those earlier years can be used to construct a plausible counterfactual for what electricity consumption would have been in year 2020 in the absence of the pandemic. This line of reasoning has been already used to show that, in most countries, official figures greatly underestimate the real death toll of the outbreak (e.g., Ciminelli and Garcia-Mandicó 2020).
Indicating with t the daily steps of our time-series, our base model is:
$$y_{t} = \beta_{0} + \mathop \sum \limits_{j = 1}^{6} \beta_{j} d_{jt} + \mathop \sum \limits_{h = 1}^{2} \beta_{h} d_{ht} + \gamma_{w} + \gamma_{w,2020}^{*} + f\left( {temp_{t} } \right) + u_{t} ,$$
(1)
where yt is the natural logarithm of electricity load, djt are six dummy variables identifying the day of the week (with Monday as the baseline), dht are two dummy variables identifying official public holidays and other observances, γw = are week-of-the-year fixed effects, γw,2020* are week-of-the-year fixed effects interacted with a dummy variable identifying year 2020, tempt = is the average air temperature and f(.) a non-linear functional form, βs are the remaining a parameters to be estimated and ut is the random component.Footnote 7
This specification is designed to capture all the peculiar features of electricity consumption and so isolate the causal impact of COVID-19. The week-of-the-year fixed effects γw encompass the slow-moving yearly seasonality of electricity load connected to both weather and cultural habits, such as the distinctive drop in consumption in the central 2 weeks of August that we mentioned in the previous section. The non-linear effect of the short-term variation in temperature is represented by f(.), which we specify as joint piecewise linear function as δ1tempt+ δ2(tempt − k)dkt, where k is the threshold in which the relationship between temperature and load reverts and δ1, δ2 are the parameters to be estimated. This flexible specification generates a V-shaped function. We set k to 62 °F (about 16.5 °C) by visually inspecting the scatterplot of the data (see Figure A1 in the Appendix in Electronic Supplementary Material). Furthermore, the effects of the weekly seasonality and the public holiday effects are captured by the corresponding dummy variables.
Our model does not include electricity price. In fact, a peculiar characteristic of electricity markets is that the demand function can be considered as completely inelastic in the short-run, since the majority of final consumers do not purchase on the wholesale exchange but, rather, are supplied by utility companies at fixed tariffs (e.g., Fezzi and Bunn 2010). These companies operate on the day-ahead market, but are required to fulfill their orders to the final consumers and, therefore, cannot respond to price variation. Because of this feature, short-run electricity load forecasting models do not typically include price information (e.g. Taylor et al. 2006) and, practically, all short-run price forecasting methods treat quantity as exogenous (e.g. Weron 2014; Fezzi and Mosetti 2020). In line with this long-standing literature, therefore, we exclude price effects from our short-run analysis of electricity consumption.
The key-parameters for our study are the coefficients γw,2020*, which measure the impact of COVID-19. These interaction effects capture the differences between each week of year 2020 and the average of the corresponding week in the previous 5 years which cannot be explained by any of the other observed factors. If our model is correctly specified, the γw,2020* parameters corresponding to the weeks before the outbreak (i.e. during January and February) should be not significantly different from zero. Of course, we still include these parameters in our model because they serve as implicit in-time placebo tests for our modelling assumptions. On the other contrary, we expect to estimate highly significant and negative γw,2020* parameters when the COVID-19 containment measures are introduced, i.e. roughly from the second week of March 2020.
If the error component ut is independently and identically distributed (iid), our model can be consistently estimated with ordinary least squares (OLS). A possible concern with this estimator is that the iid assumption may not be satisfied, since the unobserved factors represented by the stochastic component may be autocorrelated. This situation can be generated by autocorrelated measurement errors. For example, the average temperature in Milan and Rome is unlikely to perfectly represent the weather profile of the entire country and, therefore, some remaining demand variation is undoubtedly present in the error term. Since weather shocks are typically autocorrelated, this missing variation is likely to generate autocorrelation in the residuals of our model. Possible omitted variables (e.g. special events or dynamic adjustments in residential and commercial load to temperature variations) are also plausible causes of residual autocorrelation. We investigate this issue through two distinct approaches. In the first one, we apply to the OLS covariance matrix the heteroscedasticity and autocorrelation consistent (HAC) correction proposed by Newey and West (1987). By setting the maximum lag for the correlation weights at seven we also attempt to capture any remaining weekly seasonality. In the second approach, we impose an autoregressive AR(1) specification for the random component (i.e. ut = ϕut−1) and estimate the resulting model with maximum likelihood. As a further check of the robustness of our findings, we re-estimate the base model after removing the piecewise function of temperature from the equation, in order to evaluate the susceptibly of our estimates to omitted variable bias. As shown in the next section, none of these alternative specifications alter our estimates of the effect of COVID-19 lockdown in any significant way. We run our analyses in R (R Development Core Team 2006), using the packages lmtest (Hothorn et al. 2019), MASS (Ripley et al. 2013), nlme (Pinheiro et al. 2017) and sandwich (Zeileis 2004).
If our model is correctly specified and passes the time-placebo test discussed above, the coefficients γw,2020* from week 11 to week 22 can be interpreted as the causal impact of COVID-19 on electricity load. Therefore, we can estimate the impact of the pandemic and lockdown by comparing daily in-sample predictions obtained by (1) the full model and (2) a model in which such coefficients are set to zero. Indicating these two predictions (on the original scale of the variable) respectively with \(\hat{Y}_{t}\) and \(\hat{Y}_{t}^{*}\), we can write the percentage impact of COVID-19 on electricity load as:
$$l_{t} = 100\left( {\hat{Y}_{t} - \hat{Y}_{t}^{*} } \right)/ \hat{Y}_{t}^{*} ,$$
(2)
and derive appropriate confidence intervals via Monte Carlo simulations.Footnote 8
Ideally, in order to translate electricity load reductions into GDP impacts we would want to employ detailed information, disaggregated by industry type, on changes in electricity consumption, value added and amenability to distance-working solutions. Unsurprisingly, this wealth of data is not available, particularly at the daily time-scale of our analysis. As a second-best solution, we employ some deliberately simple and intuitive assumptions to transform our estimates of electricity load changes into GDP impacts.
We assume that, in the short run and at the national level, GDP changes are proportional to the changes in electricity consumption by all productive sectors (i.e. all sectors but the residential one). In order to evaluate this claim, we run a back-of-the-envelope calculation on GDP and electricity consumption information for Italy for the years 1990–2018.Footnote 9 Both variables are non-stationary and, therefore, we compute a correlation analysis on the first differences (in percentage) in order to avoid measuring a spurious relation. We estimate a correlation coefficient of 0.88, indicating an extremely strong linear covariation in the long-run. We believe this relationship to be even stronger in the short-run, justifying the simple assumption of a 1:1 relationship in our calculations (the plot is reported in Figure A2 in the Appendix in Electronic Supplementary Material).
In order to derive the impact of COVID-19 on the electricity consumption of the productive sectors we need to rescale our estimates, which are calculated on the total load. We compare two simple approaches. In the first one, we assume that residential consumption has remained unaffected by the restrictions and, therefore, all the reduction in electricity load due to the COVID-19 can be traced back to the other sectors. In the second one, we follow recent International Energy Agency’s estimates (IEA 2020) reporting that residential consumption has increased by 40% during the lockdown, and rescale our calculations accordingly. The percentage GDP impacts following these two methods can be written as:
$$GDP_{{ 1 {\text{t}}}} = l_{t} 100/\left( { 100{-}r} \right),\,{\text{and}}$$
(3)
$$GDP_{{ 2 {\text{t}}}} = l_{t} 100/\left( { 100{-} 1. 4r} \right),$$
(4)
where r represents the percentage of consumption of the residential sector, which in Italy corresponds to 22.4% according to the most up-to-date IEA estimates (footnote 3). While for transparency we report both measures, we believe that (4) should be the preferred estimate during the lockdown months (March and April) while (3) should be more accurate for the post-lockdown period, i.e. from the month of May onward. Although extremely simple, the next section shows that these assumptions provide results that are remarkably close to the official estimates of the GDP changes during the first quarter of 2020. Of course, our results extend well beyond the first quarter and provide an up-to-date assessment of the status of the economic disruption caused by the pandemic.