4 Methodology and Research Design
4.1 Empirical strategy
This study adopts the ARDL bounds testing approach. Unlike static cointegration techniques, the ARDL framework accommodates regressors with mixed orders of integration (Pesaran et al., 2001). Since the focus is the recent transition period, the approach also has the advantage of performing acceptably in smaller samples (Wolde-Rufael, 2010). This flexibility is appropriate for Taiwan’s energy transition, where policy interventions may introduce structural breaks and varying stationarity properties across series.
4.2 Long-run specification
A log-linear functional form is specified so that coefficients read as elasticities. The long-run relationship is
\[ \ln(FFI^{power}_t) = \alpha_0 + \alpha_1 \ln(RE_t) + \alpha_2 \ln(EC_t) + \alpha_3 NUC^{share}_t + \alpha_4 \ln(ER_t) + \alpha_5 \ln(PF_t) + \psi D_t + \phi B_t + \epsilon_t , \]
where \(D_t\) is a vector of eleven centered monthly dummies absorbing seasonality and \(B_t\) is a break dummy switching on in November 2022. Two elements depart from the original research proposal: nuclear power enters as a share rather than in logarithms, for the reason given in the previous chapter, and the fuel price index is added. Both departures are motivated by the data rather than by convenience, and both are subjected to robustness checks.
Seasonality is not incidental. Wind generation runs 90% above its mean in December and 60% below in August, solar runs in the opposite direction, and both \(FFI^{power}\) and electricity consumption peak in the summer, reflecting the cooling-demand surge documented for Taiwan by Hung & Huang (2015) and Lan & Lee (2011) and the counter-cyclical wind profile reported by Taipower (Taiwan Power Company (Taipower), 2025). Without seasonal controls these opposing cycles would generate spurious correlation.
4.3 Unrestricted Error Correction Model
The long-run equation is re-parameterized into an Unrestricted Error Correction Model:
\[ \begin{aligned} \Delta \ln(FFI_t) &= \beta_0 + \sum_{i=1}^{p-1} \beta_{1i} \Delta \ln(FFI_{t-i}) + \sum_{i=0}^{q_1-1} \beta_{2i} \Delta \ln(RE_{t-i}) \\ &\quad + \sum_{i=0}^{q_2-1} \beta_{3i} \Delta \ln(EC_{t-i}) + \sum_{i=0}^{q_3-1} \beta_{4i} \Delta NUC^{share}_{t-i} + \sum_{i=0}^{q_4-1} \beta_{5i} \Delta \ln(ER_{t-i}) \\ &\quad + \sum_{i=0}^{q_5-1} \beta_{6i} \Delta \ln(PF_{t-i}) + \lambda_1 \ln(FFI_{t-1}) + \lambda_2 \ln(RE_{t-1}) + \lambda_3 \ln(EC_{t-1}) \\ &\quad + \lambda_4 NUC^{share}_{t-1} + \lambda_5 \ln(ER_{t-1}) + \lambda_6 \ln(PF_{t-1}) + \psi D_t + \phi B_t + \mu_t . \end{aligned} \]
The \(\beta\) coefficients capture short-run dynamics and the \(\lambda\) coefficients on the lagged levels embody the long-run relationship. Lag orders are selected by the Akaike Information Criterion, subject to a maximum of two and to the selected model passing a Ljung–Box test for serial correlation at twelve lags. The maximum is deliberately restrictive: with 121 monthly observations in the transition sample and eleven seasonal dummies, higher orders would exhaust the degrees of freedom.
4.4 Cointegration testing
Testing proceeds in three stages rather than one. The bounds test examines the joint restriction \(H_0: \lambda_1 = \cdots = \lambda_6 = 0\); rejection against the upper critical bound indicates a stable long-run relationship (Pesaran et al., 2001). Because the asymptotic critical values over-reject in small samples, critical values are simulated at the actual sample size with 100,000 replications, calibrated to the exact sample size and regressor count used here, in the spirit of the small-sample tabulations of Narayan (2005).
A significant overall \(F\) statistic is, however, not sufficient, because degenerate cases can produce it. The three-test protocol of McNown et al. (2018) is therefore applied: cointegration requires rejecting the overall \(F\) test, the \(t\) test on the lagged dependent variable, and the \(F\) test on the lagged independent levels jointly. All three must reject.
4.5 Elasticity and inference
Conditional on cointegration, the long-run elasticity of renewable energy is recovered by normalizing the lagged level coefficients:
\[ \eta_{RE} = - \frac{\lambda_2}{\lambda_1} . \]
A statistically significant negative \(\eta_{RE}\) validates the substitution hypothesis; a value approaching zero supports the additionality paradox.
Because \(\eta_{RE}\) is a ratio of estimates, its finite-sample distribution is skewed and delta-method standard errors can understate the tails. Confidence intervals are therefore reported both from the delta method and from a residual bootstrap with 2,000 resamples (Efron, 1979), with the bootstrap interval preferred.
Stationarity is assessed with the Augmented Dickey–Fuller test, the KPSS test (Kwiatkowski et al., 1992), and the Zivot–Andrews test allowing one endogenous break (Zivot & Andrews, 1992). Parameter stability is assessed with recursive-residual CUSUM and CUSUMSQ statistics (Brown et al., 1975). All estimation is carried out in Python using statsmodels (Seabold & Perktold, 2010), with figures produced using seaborn (Waskom, 2021).