\[\def\E{\mathbb{E}} \def\Var{\operatorname{Var}} \def\Cov{\operatorname{Cov}}\]
Complete both problems. All observations are in the accompanying data folder; the links below open the individual CSV files. Write the reasoning by hand and append labeled computational exhibits, following the Homework 5 assignment sheet. No additional data collection is required.
Problem 1. Investigating Stationary Relations
The four pairs below cover companies, crops, shipping routes, and an unusual currency predictor. For this problem, take natural logarithms of both recorded series. In each regression, \(y_t\) is the response and \(x_t\) the predictor specified here.
| Pair | \(x_t\) | \(y_t\) | Observations | Bartlett bandwidth \(H\) |
|---|---|---|---|---|
| Coca-Cola / PepsiCo | log Coca-Cola | log PepsiCo | 313 weekly, 2018–2023 | 4 weeks |
| Wheat / corn | log corn | log wheat | 384 monthly, 1992–2023 | 5 months |
| Grain freight | log Pacific Northwest rate | log Gulf rate | 336 monthly, 1996–2023 | 5 months |
| Cheese / yen | log cheese production | log yen per dollar | 624 monthly, 1972–2023 | 5 months |
Let \(X_t,Y_t\) denote the log-level processes. For these empirical comparisons, allow a deterministic linear drift in addition to the \(I(1)\) component defined in Workbook 10: write \(X_t=\delta_Xt+S_t\) and \(Y_t=\delta_Yt+R_t\), where \(S_t,R_t\) are \(I(1)\) under that definition. A stationary spread \(Y_t-a-bX_t\) requires the linear drifts to cancel as well: \(\delta_Y=b\delta_X\). Level-KPSS on first differences assesses stationarity around an unknown constant; it does not test whether that constant is zero.
Assess plausibility. Before fitting or testing, read the descriptions below. For each pair, explain in two or three sentences why a cointegrating relationship seems plausible or implausible, based on what the series measure. Describe what could link their long-run movements, or why there is little reason to expect a stable relation.
Examine the individual series. Using the supplied plots and your plots of the first log differences \(\nabla x_t=x_t-x_{t-1}\) and \(\nabla y_t=y_t-y_{t-1}\), assess whether the log levels are plausibly described by the integrated model above. Apply level-KPSS to each log level and each first log difference using the supplied Bartlett bandwidth. State the null and use the 5% cutoff \(0.463\). Give one reason these checks do not establish every assumption of an \(I(1)\) model.
Fit the relation. Fit \(y_t=a+bx_t+u_t\) by OLS with an intercept. Report \(\widehat a\), \(\widehat b\), and \(R^2\). Plot the fitted residuals against time and their sample ACVF through lag 40. Do their level, size, and dependence support a stationary approximation?
Examine the residual KPSS statistic. Compute the KPSS statistic for the fitted residuals using the same Bartlett bandwidth. Report the statistic and its long-run variance estimate. Explain why the ordinary KPSS critical values and package p-values do not calibrate a test of this estimated relation. Interpret the numbers together with the plots in (c).
Make the case. For each pair, write three or four sentences giving the strongest evidence for a stationary relation, the strongest concern, and a tentative conclusion. Separate the evidence about the individual levels from the evidence about their combination. State whether the diagnostics changed your assessment in (a), and why. Explain why either a high \(R^2\) or an implausible scientific story is insufficient to settle cointegration.
Submit one compact display per pair containing the first log differences, the fitted residual series, and its sample ACVF. Combine the fitted coefficients and diagnostic results in tables. The same calculation can be reused across the four datasets; the interpretations should refer to their different behavior. Workbook 9 gives the Bartlett-normalized KPSS statistic; Workbook 10, Estimating and Assessing a Relation, explains the difference between a fixed and an estimated spread.
Coca-Cola and PepsiCo
Coca-Cola and PepsiCo both sell branded beverages; PepsiCo also has a substantial food business. Their share prices can respond to common consumer demand and market conditions while their businesses develop differently. The file contains 313 weekly observations, January 5, 2018, through December 29, 2023. Prices are Yahoo Finance adjusted closes, adjusted for splits and cash dividends, in the vendor’s adjusted dollars-per-share scale.
Download the observations. Use coca_cola_adjusted_usd as the predictor and pepsi_adjusted_usd as the response before taking logs. Each date is the final common trading session of its week.
Wheat and Corn
Wheat can substitute for corn in livestock feed. This creates a reason for their prices to respond to each other, while harvest conditions and other uses can affect them differently. The observations are monthly global benchmark prices, in nominal U.S. dollars per metric ton, from January 1992 through December 2023.
Download the observations. Use corn_usd_per_metric_ton as the predictor and wheat_usd_per_metric_ton as the response before taking logs. The date identifies the month.
Shipping Grain to Japan
These are ocean freight rates for bulk grain shipped to Japan from the U.S. Gulf and from the Pacific Northwest. The cargo category and destination are shared; the routes and voyage lengths differ. The file contains monthly rates in U.S. dollars per metric ton, January 1996 through December 2023, published in USDA’s Grain Transportation Report.
Download the observations. Use pnw_usd_per_metric_ton as the predictor and gulf_usd_per_metric_ton as the response before taking logs. The date identifies the month.
USDA dataset description · Source spreadsheet. Seven source dates for June–December 2019 carry the year 1919; the supplied CSV corrects the year using their position in the consecutive monthly record.
Cheese Production and the Yen
A hypothetical currency analyst proposes using U.S. cheese production to value the Japanese yen. The observations pair a seasonally adjusted cheese-production index, with 2017 equal to 100, and the number of yen per U.S. dollar. There are 624 monthly observations, January 1972 through December 2023. A rise in yen per dollar means that the yen buys fewer dollars.
Download the observations. Use cheese_production_index as the predictor and yen_per_usd as the response before taking logs. Cheese is a production-volume index, not a price; the exchange rate is the monthly arithmetic average of available daily observations.
Problem 2. Deriving a Pairs Trade
A hedge fund is a private investment fund that pools investors’ money and uses strategies that can include borrowing and short selling. In this hypothetical assignment, the fund specializes in market-neutral long–short strategies: it simultaneously buys shares of one company and sells borrowed shares of another in the same industry, aiming to reduce exposure to market-wide price movements.
The fund asks for a trading rule for WEC Energy Group and Xcel Energy. Both operate electric and natural-gas utilities, with substantial operations in the Midwest. Their prices may respond to common changes in financing costs, energy demand, and regulation. The firms remain distinct businesses, so such a connection is a reason to investigate their relative prices.
The training file contains 209 weekly closing prices from 2018–2021. Let \(W_t\) and \(X_t\) denote WEC and Xcel prices in dollars per share. This problem uses dollar prices directly. The recorded closes are split adjusted but are not adjusted for cash dividends; there were no stock splits during the supplied period.
OLS on the training observations, followed by an AR(1) fit to its gap, gives the following working model: \[ W_t=a+bX_t+U_t,\qquad U_{t+1}=\phi U_t+Z_{t+1}, \] with \[ a=1.146277,\qquad b=1.385665,\qquad \phi=0.876663. \] For this problem, assume \(W_t\) and \(X_t\) share an integrated component and \(U_t\) is the stationary solution of the displayed AR(1) equation. For every \(t\), the innovation \(Z_{t+1}\) has mean zero, finite positive variance, and is independent of the joint price history \(\mathcal H_t=((W_s,X_s):s\le t)\). Treat the displayed estimates as fixed model parameters throughout; do not refit on the later observations.
The analyst can calculate the current gap \(u_t=w_t-a-bx_t\) from the observed prices. The fund has supplied no model for the future common movement of the two stocks. The question is whether the gap model is enough to design a position whose expected price gain can be calculated.
- Forecast the gap. Derive \(\E[U_{t+h}\mid\mathcal H_t]\) and the expected change \(\E[U_{t+h}-U_t\mid\mathcal H_t]\). For \(h=1,4,13\), calculate the fraction of the current gap that the model predicts will close. What does the AR(1) assumption add to the assertion of cointegration?
A position of \(q\) shares earns the price gain \(q(p_{t+h}-p_t)\) when held from \(t\) to \(t+h\). A negative \(q\) is a short position: it gains when the price falls. Fractional shares and short positions are allowed in this exercise. Holdings remain fixed during a trade. A cash account finances the initial stock position at zero interest.
Explain the trading idea. In a few sentences, explain how the cointegrating relation and the gap forecast could guide the fund’s long–short strategy. Describe which stock the fund would buy and which it would sell short when the gap is positive, and how the strategy would change when it is negative. Explain why combining these positions could reduce exposure to their shared price movements. Give the idea without calculating share counts.
Choose the shares. Hold exactly 100 WEC shares in absolute value. Choose the number of Xcel shares so that the combined price gain depends only on the change in \(U\), for any changes in the individual prices. Derive the WEC and Xcel positions that maximize conditional expected price gain when \(u_t>0\) and when \(u_t<0\). If \(u_t=0\), take no position. Give the required initial cash-account balance. Express the realized price gain in terms of \(u_t\) and \(u_{t+h}\), and its conditional expectation using the current gap and model parameters.
Write down the rule from (c) before opening the evaluation file. It contains the last training observation at \(t=0\), December 31, 2021, followed by 104 weekly observations from 2022–2023. Enter at \(t=0,13,26,39,52,65,78,91\) and close each position at \(t+13\). At each entry, use the prices then observed and the same fixed coefficients. For this idealized calculation, assume the observed prices are immediately available for execution. Close the old position before opening the next one at a shared boundary.
Calculate price gains only, ignoring dividends, interest, and trading costs.
Use the later observations. For each of the eight trades, report the entry and exit dates, entry gap, both share positions, predicted price gain, and realized price gain. Report the totals and the number of trades with a negative realized price gain. Keep the prescribed dates and the rule from (c).
Assess the strategy. Explain why positive conditional expected price gain need not produce a positive realized gain on every trade. What information does the current gap make directly available that a model using only recent price differences might omit? Explain whether keeping all differences and both initial prices would lose that information. Identify one feature of actual trading omitted from this calculation and how it could affect the reported gain.
Prices: Yahoo Finance WEC and XEL, last common trading session of each week. Company context: WEC Energy Group and Xcel Energy. Background: Hedge funds, Investor.gov.