Explainable Artificial Intelligence (XAI) is increasingly required in computational economics, where machine-learning forecasters can outperform classical econometric models but remain difficult to audit and use for policy. This survey reviews and organizes the growing literature on XAI for economic time series, where autocorrelation, non-stationarity, seasonality, mixed frequencies, and regime shifts can make standard explanation techniques unreliable or economically implausible. We propose a taxonomy that classifies methods by (i) explanation mechanism: propagation-based approaches (e.g., Integrated Gradients, Layer-wise Relevance Propagation), perturbation and game-theoretic attribution (e.g., permutation importance, LIME, SHAP), and function-based global tools (e.g., Accumulated Local Effects); (ii) time-series compatibility, including preservation of temporal dependence, stability over time, and respect for data-generating constraints. We synthesize time-series-specific adaptations such as vector- and window-based formulations (e.g., Vector SHAP, WindowSHAP) that reduce lag fragmentation and computational cost while improving interpretability. We also connect explainability to causal inference and policy analysis through interventional attributions (Causal Shapley values) and constrained counterfactual reasoning. Finally, we discuss intrinsically interpretable architectures (notably attention-based transformers) and provide guidance for decision-grade applications such as nowcasting, stress testing, and regime monitoring, emphasizing attribution uncertainty and explanation dynamics as indicators of structural change.