## 3 Prediction and Future of EVs in the UK
### Mathematical Formulation of Models
#### 3.3.2 SARIMAX Model  
The SARIMAX model adds autoregressive and moving average terms to a regression on exogenous variables:  
\[y_{t}=\beta_{0}+\sum_{i=1}^{p}\varphi_{i}y_{t-i}+\sum_{j=1}^{q}\theta_{j} \epsilon_{t-j}+\sum_{k=1}^{K}\beta_{k}x_{k,t}+\epsilon_{t} \tag{3}\]  
where:  
\(y_{t}\): is the value at time t (EV sales);  
\(\beta_{0}\): is the constant term;  
\(p\): refers to the number of autoregressive (AR) terms. These terms represent past values of the dependent variable \(y_{t}\), which are included to account for its own temporal dependence;  
\(q\): denotes the number of moving average (MA) terms. The MA terms represent past error terms (residuals) that are included to correct for autocorrelated shocks in the time series;  
\(K\): is the number of exogenous regressors--external predictor variables that are believed to influence \(y_{t}\), such as battery price, charger density, GDP growth, and policy indicators;  
\(\varphi\): is the coefficient for the \(i\)-th AR term;  
\(\theta_{j}\): is the coefficient for the \(j\)-th MA term;  
\(\beta_{k}\): is the coefficient for the \(k\)-th exogenous regressor \(x_{k,t}\);  
In this study, the SARIMAX model was specified with one autoregressive lag (\(p=1\)) and no moving average component (\(q=0\)). This decision was guided by the short time span of available data, just ten annual observations from 2015 to 2024, which limits the degrees of freedom and reduces the reliability of more complex time-series structures. Introducing additional AR or MA terms would increase model complexity and the risk of overfitting, especially in such a small-sample setting. Moreover, the inclusion of four (K = 4) informative exogenous regressors--battery price, charger density, GDP growth, and policy dummy--captured the majority of the explainable variance in EV sales, as indicated by the model's high in-sample performance (R\({}^{2}\) = 0.978 as listed below in Table 3). The SARIMAX(1,0,0) specification therefore offered a strong balance between parsimony, interpretability, and forecasting accuracy and was selected as the most robust option for this analysis.  
Additionally, the model was estimated without an explicit intercept term (\(\beta_{0}\)). This was appropriate because time-dependent and level-setting regressors--particularly charger density, battery price, and the policy indicator had already captured the underlying trend and scale of EV sales over the study period. In such configurations, the intercept becomes statistically redundant and is therefore omitted by standard estimation procedures when exogenous variables are present. Importantly, the exclusion of \(\beta_{0}\) did not compromise model performance: the specification maintained a high in-sample fit (R\({}^{2}\) = 0.978) and low residual variance (\(\sigma^{2}\approx 692\)). The absence of an intercept does not introduce bias in this case, as the included regressors collectively represent the structural dynamics of the target variable. This reduced form yields a stable, interpretable, and well-specified model, particularly suited to the short historical series. Accordingly, Equation (3) can be expressed in its reduced form without an intercept:  
\[y_{t}=\varphi_{1}y_{t-1}+\sum_{k=1}^{4}\beta_{k}x_{k,t}+e_{t} \tag{4}\]  
where:  
\(\varphi_{1}\): is the autoregressive coefficient (only one lag used in SARIMAX(1,0,0));  
\(x_{\text{kt}}\): are the exogenous regressors at time \(t\);  
\(\epsilon_{\text{t}}\): is the error term.