Explain the three components of an ARIMA(p, d, q) model. Then describe how you use the ACF and PACF of a series to choose the orders p and q, and how you decide on d. Finish by saying what you would typically expect for a series of liquid asset returns.

The three componentsARIMA(p, d, q) combines: AR(p) — autoregression: the value is a linear function of its own $p$ past values, $x_t = c + \sum_{i=1}^{p}\phi_i x_{t-i} + \epsilon_t$. I(d) — integration: difference the series $d$ times to make it stationary before modelling. MA(q) — moving average: the value depends on the last $q$ shock terms, $x_t = c + \epsilon_t + \sum_{j=1}^{q}\theta_j \epsilon_{t-j}$. Choosing d$d$ is about stationarity, not fit. Difference the series until it is stationary, judged by eye (no trend, stable variance) and by an ADF test. The tell-tale sign that you still need to difference is an ACF that decays very slowly, almost linearly, instead of dropping off. Most financial series need $d = 1$ (prices to returns); over-differencing is a real mistake and shows up as a strong negative spike at lag 1. Choosing p and q from ACF/PACFAfter differencing, read the two plots: ModelACFPACF AR(p)decays / tails offcuts off after lag p MA(q)cuts off after lag qdecays / tails off ARMA(p,q)tails offtails off The intuition: the PACF isolates the direct effect of lag $k$ after removing the intermediate lags, so a pure AR(p) has exactly $p$ nonzero partials. The ACF includes indirect effects, so a pure MA(q) — which truly only depends on $q$ shocks — cuts off cleanly in the ACF. Then confirmCandidate orders are starting points; fit by maximum likelihood, compare models by AIC/BIC, and check that the residuals look like white noise (flat ACF, Ljung-Box test). What to expect for liquid returnsEfficient, liquid markets make returns close to white noise, so the ACF and PACF are mostly inside the confidence bands and the best ARIMA is often trivial ($p, q$ small or zero). The interesting structure in finance is usually in the variance, not the mean — which is what GARCH models, not ARIMA, are for.

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def two_sum(nums, target):
    seen = {}
    for i, n in enumerate(nums):
        if target - n in seen:
            return [seen[target - n], i]
        seen[n] = i
    return []

print(two_sum([2, 7, 11, 15], 9))  # [0, 1]

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The Interview Guide for Quants and Traders

Identifying an ARIMA Model

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