Following previous literature (e.g. Sailor and Mu˜noz, 1997, Considine, 2000, and Silk and Joutz, 1997), I use space conditioning variables to estimate the effect of weather on electricity demand. I analyze the impact of cooling and heating degree days on the demand for electricity assuming that the data generating process is given by equation 2.1 and estimate the electricity model for each end-use sector i,
with i = (residential, commercial, industrial), according to equation 2.2:
E∗= β0eβcddCDDeβhddHDDP EβP EIβI (2.1)
ln Ei,s,t = β0,i+ βcdd,iCDDs,t+ βhdd,iHDDs,t+ βP E,iln P Ei,s,t+ βI,i ln Ii,s,t+ µi,s,t
(2.2) in which, Ei,s,trepresents the quantity of electricity consumed by sector i in state s at time
t (in GWh). CDDs,t and HDDs,t are monthly cooling degree days and heating degree
days, respectively, in state s at time t. Since both CDDs,t and HDDs,tare constructed at
the climatic region level, I aggregate them by computing a weighted average for each state. As weights I use the percentage of the total population in each climatic region. Given that monthly CDDs,t and HDDs,t time series can take the value of zero during summer and
winter seasons respectively (and therefore are not included in the equation 2.2 in logs), the coefficients βcdd,i and βhdd,i have to be interpreted as semi-elasticities with respect to
in state s at time t. Ii,s,t represents either personal income or employment. Income is
used to estimate the model for the residential sector, and sectoral employment is used to estimate the model for both commercial and industrial demands. These employment variables are proxies for the economic activity in these sectors.10 Given that E, P E and I are transformed into natural logarithms, elasticities of electricity demand can be directly obtained and interpreted. Finally, µi,s,t represents the stochastic error term.
I expect the coefficients associated with CDD, HDD, income and employment to be positive, while the coefficient associated with electricity price to be negative. I also expect the coefficient on CDD (βcdd) to be greater than the coefficient on HDD (βhdd) since annual
peak demand occurs during summer months when there is an intensive use of electrical appliances for space cooling. I estimate equation 2.2 by implementing the following models: 1) state fixed effects model, 2) DOLS model at the state level when I fail to reject the existence of unit roots in the time series, and 3) OLS model at the state level for those states that reject the existence of unit roots. These three models are explained in the following sections.
2.3.1 State Fixed Effects Model
I estimate this model to compare our results with previous estimates of the effect of degree days on demand. In this model, heterogeneity across states is captured by differences in the constant term. A different intercept is estimated for each state by adding a set of forty-
10
A direct measure of the economic activity is the GDP by sector. However, GPD-by-state data disag- gregated among industries are available only at an annual frequency. A second problem with constructing a single GDP time series from 1990 to 2010 is that there is a discontinuity in 1997, where the industry classification changed from SIC to NAICS. Given that NAICS-based GDP-by-state estimates are consis- tent with U.S. gross domestic product (GDP) and SIC-based GDP-by-state estimates are consistent with U.S. gross domestic income (GDI), BEA strongly cautions against appending these two data series in an attempt to construct a single time series of GDP-by-state.
seven dummies ηsto equation equation 2.2. The state of Alabama is considered as reference
and its intercept corresponds to β0,i. The constant term of any other state s is given by β0,i
+ ηs. Note that coefficients on CDD and HDD (βcdd and βhdd) are constrained to be the
same across states. In addition, annual dummies (θy) and seasonal (quarterly) dummies
(φq) are included in equation 2.2 to take into account different patterns of electricity
consumption across years and seasons. I include three seasonal dummies (spring, summer and f all), and use winter as the omitted category.
2.3.2 Unit Roots, Seasonal Cointegration and DOLS Estimation
To better assess the heterogeneity across states I allow for differences in both intercepts and slope coefficients. In this model, I estimate one equation for each state s and for each end-use sector i. Thus, a different constant term β0,s and different coefficients on CDD
and HDD (βcdd,s and βhdd,s) are estimated for each state s.
Time series for electricity demand and its determinants can contain a stochastic trend, in which case the estimation of a statistically significant relationship among non-stationary variables would lead to a spurious regression. I use the MHEGY procedure developed by Beaulieu and Miron (1993) to test for unit roots in all variables of equation 2.2 (electricity demand, CDD, HDD, electricity price, income and employment). This procedure performs a HEGY test (Hylleberg et al., 1990) for seasonal unit roots when analyzing monthly data. The null hypothesis of the test is that the variable contains a seasonal unit root, and the alternative is that the variable is generated by a stationary process. I perform the test for all frequencies, being the annual frequency, or the zero frequency (π = 0), the one of interest for this analysis. Specifically, I perform the MHEGY test including a constant, a linear time trend and a deterministic seasonal for electricity demand and electricity price.
In the case of CDD and HDD the test includes a constant and a deterministic seasonal. Finally, I test for income and employment using a constant and a linear trend. I obtain the critical values for the distributions of t-statistics for seasonal unit roots from Beaulieu and Miron (1993). In the presence of seasonal unit roots, electricity demand can share a common stochastic trend with its determinants, in which case time series are said to be cointegrated. Following Engle and Granger (1987), I perform the unit root test on the residuals of equation 2.2 estimated by OLS. Thus, I analyze the error term (µi,s,t)
using the ADF test (Dickey and Fuller, 1979) and choosing the lag length with the Akaike Information Criterion, AIC (Akaike, 1973). If the error term is I(0), time series from the demand model are said to be co-integrated.
Following Stock and Watson (1993), I estimate the cointegrating relationship among non-stationary variables of equation 2.2 using dynamic ordinary least squares (DOLS). DOLS generates asymptotically efficient estimates of the regression coefficients which cor- respond to the long run relationship among variables. I choose leads and lags of the first difference of significant variables using the Schwarz Information Criterion (Schwarz, 1978). Given that the set of variables that fails to reject the null hypothesis of unit root will differ across states and end-use sectors, I select the following three models on which I perform the cointegration test: (a) electricity demand as a function CDD, HDD, electric- ity price and income/employment; (b) electricity demand as a function CDD, HDD and income/employment; and (c) electricity demand as a function CDD, HDD and electricity price. To select the most appropriate model for each state, I estimate the DOLS model in those states and end-use sectors in which I find evidence of cointegration. I select the model with the highest R2 in those cases in which there is more than one DOLS model
for the same state and same type of consumption.
Finally, I estimate an error correction model (ECM) to analyze the short-run dynam- ics of the cointegrating relationship. The ECM in first differences includes lags of both dependent and independent variables. Given that all variables in first differences are sta- tionary, I perform the ECM by OLS. I choose the lag length using the AIC criterion. The error term corresponds to the difference between the dependent variable of the DOLS es- timation (ln Ei,s,t) and the coefficients βcdd,i,s, βhdd,i,s, βpe,i,s, and βI,i,s multiplied by the
independent variables CDDs,t, HDDs,t, P Ei,s,t, and It, respectively. The magnitude of
its coefficient represents the rate at which electricity demand responds to disequilibrium in the cointegrating relationship, while its sign represents whether electricity demand moves toward the equilibrium value implied by the long-run relationship. Thus, a negative value is desired because it indicates that the disequilibrium between electricity demand and the right-hand side variables of the DOLS estimation causes electricity demand to move toward the long-run equilibrium that is implied by the right hand side variables on the cointegrating relationship.
2.3.3 OLS Model by State
In those states in which time series reject the existence of unit roots, I estimate the relationship between electricity demand, cooling degree days, electricity price and income by OLS. As explained in Section 2.3.2 I allow for differences in both intercepts and slope coefficients for each state s and for each end-use sector i. I include both annual (θy)
and seasonal (quarterly) dummies φq in equation 2.2 to control for seasonal and annual
variations in electricity consumption. I also include three dummies (spring, summer and fall) and use winter as the omitted category. Therefore, the estimates of the coefficients
on CDD and HDD measure the effect of weather on electricity demand beyond changes across season. That is, βcdd,sand βhdd,s measure the effect of temperature in months that
are hotter or cooler than a normal season.