The following graph displays a monthly time series of the exchange rate between the US dollar and the Euro, available on Canvas (exchange_usd_eur.csv). The sample covers the period between January 1999 and September 2024. Define the training sample as all the information available up to December 2022.

Assume that the exchange rate data follows an process in differences. More specifically,

where and . What is the estimate of ?

Question

The following graph displays a monthly time series of the exchange rate between the US dollar and the Euro, available on Canvas (exchange_usd_eur.csv). The sample covers the period between January 1999 and September 2024. Define the training sample as all the information available up to December 2022.

Assume that the exchange rate data follows an  process in differences. More specifically,

where  and . What is the estimate of ?

BlackTom AI · Accepted Answer

First, the task asks us to model the exchange rate data in differences using a simple autoregressive process of order 1 on the differences. In time series terms, if y_t denotes the USD/EUR exchange rate, then we consider the difference Δy_t = y_t − y_{t−1}. The assumed model is: Δy_t = φ Δy_{t−1} + ε_t, where ε_t is a white noise error term.

Next, to estimate φ, we perform ordinary least squares (OLS) regression of the observed Δy_t on the lagged difference Δy_{t−1} using the training sample (all data up to December 2022). The procedure is straightforward: compute the successive differences Δy_t from the series, align them with their previous values Δy_{t−1}, and fit the regression Δy_t ~ Δy_{t−1}. The estimated coefficient on Δy_{t−1} is the estimate of φ.

Interpretation: φ measures the persistence in changes of the exchange rate. If φ ≈ 0, changes are largely random from month to month; if φ > 0, positive changes tend to be followed by positive changes and negative changes by negative changes, with larger φ implying stronger momentum in the moves. An estimate around 0.3 would indicate a modest positive autocorrelation in monthly changes, i.e., some persistence but not strong momentum.

Practical considerations: the graph spans 1999–2024 with potential structural breaks (crises, policy shifts) and possible nonstationarity in levels. When working with Δy_t, these issues are mitigated to some extent, but you should still check residual diagnostics for autocorrelation, heteroskedasticity, and potential regime changes. The sample up to December 2022 provides the data points used to compute φ; data beyond that date are not included in the estimation.

If the reported answer is 0.3, it would reflect the OLS estimate of the coefficient on Δy_{t−1} from that regression on the training sample, consistent with the described procedure and interpretation.

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