Chapter 23 Quiz: Time Series Analysis for Betting

Instructions: Answer all 25 questions. This quiz is worth 100 points. You have 75 minutes. A calculator is permitted; no notes or internet access. For multiple choice, select the single best answer.

Section 1: Multiple Choice (10 questions, 3 points each = 30 points)

Question 1. A time series is said to be weakly stationary if:

(A) Its values never change over time

(B) Its mean, variance, and autocovariance are constant over time

(C) It has no autocorrelation at any lag

(D) It follows a normal distribution at every time point

Question 2. The null hypothesis of the Augmented Dickey-Fuller (ADF) test is:

(A) The series is stationary

(B) The series has a unit root (non-stationary)

(C) The series has no autocorrelation

(D) The series has constant variance

Question 3. In an ARIMA(2,1,1) model, what does the "1" in the middle position represent?

(A) One autoregressive lag

(B) One moving average term

(C) The series has been differenced once to achieve stationarity

(D) The model has one seasonal component

Question 4. A team's weekly scoring margin shows a significant positive autocorrelation at lag 1 of 0.45. This most likely indicates:

(A) The team's performance is purely random from week to week

(B) Good performances tend to be followed by good performances (momentum/persistence)

(C) The team mean-reverts rapidly after each game

(D) The data contains a seasonal pattern with period 1

Question 5. The half-life of a mean-reverting process with speed parameter $\theta = 0.10$ is approximately:

(A) 2.3 time periods

(B) 4.6 time periods

(C) 6.9 time periods

(D) 10.0 time periods

Question 6. Which changepoint detection algorithm is specifically designed for online (real-time) detection rather than retrospective analysis?

(A) PELT (Pruned Exact Linear Time)

(B) Binary segmentation

(C) BOCPD (Bayesian Online Changepoint Detection)

(D) CUSUM (Cumulative Sum)

Question 7. An NBA team goes 8-2 in their first 10 games, then 4-6 in their next 10 games. Which of the following is the BEST interpretation from a time series perspective?

(A) The team has definitely declined in quality

(B) This pattern is consistent with random variation around a true 60% win rate

(C) This is clear evidence of mean reversion

(D) A changepoint has occurred between games 10 and 11

Question 8. In the context of time series analysis for betting, "closing line value" (CLV) measured over time is best analyzed using:

(A) A single-point hypothesis test

(B) Time series methods to detect whether the edge is persistent, trending, or mean-reverting

(C) A simple win-loss record

(D) Cross-sectional regression with no temporal component

Question 9. Which of the following is NOT a valid approach to handling non-stationarity in a sports time series?

(A) First differencing to remove a linear trend

(B) Seasonal differencing to remove periodic patterns

(C) Log transformation to stabilize variance

(D) Removing all data points that are more than two standard deviations from the mean

Question 10. A SARIMA(1,0,0)(1,0,0)[82] model for NBA team performance includes a seasonal AR term at lag 82. This term captures:

(A) The effect of playing the same team again in the same season

(B) The tendency for performance at the same point in consecutive seasons to be correlated

(C) Weekly patterns within a single season

(D) The influence of the All-Star break on performance

Section 2: True/False (5 questions, 3 points each = 15 points)

Question 11. True or False: If a time series has a significant trend, an ARMA model (without differencing) fitted to the raw data will produce biased and unreliable forecasts.

Question 12. True or False: Mean reversion in sports statistics implies that after a period of unusually good performance, a team's future performance will be worse than their historical average.

Question 13. True or False: The PELT algorithm for changepoint detection guarantees finding the globally optimal set of changepoints under its cost function.

Question 14. True or False: If the ACF of a time series decays slowly and the PACF has a sharp cutoff after lag 2, the appropriate model is MA(2).

Question 15. True or False: A calendar effect (e.g., poor NFL team performance on Thursday Night Football) that has been widely publicized and known to bettors for several years is unlikely to still offer a profitable betting edge.

Section 3: Fill in the Blank (3 questions, 4 points each = 12 points)

Question 16. In an Ornstein-Uhlenbeck process, the parameter $\theta$ controls the _ of mean reversion. A larger value of $\theta$ means the process reverts to its long-run mean _ quickly.

Question 17. The Akaike Information Criterion (AIC) balances model _ (measured by the likelihood function) against model _ (measured by the number of parameters). Lower AIC values indicate a ____ model.

Question 18. In CUSUM changepoint detection, the cumulative sum statistic is computed as $S_k = \sum_{i=1}^{k}(x_i - \bar{x})$. The estimated changepoint is the value of $k$ at which $|S_k|$ is _. Before the changepoint, the series mean is _ the overall mean, and after the changepoint, the series mean is ____ the overall mean.

Section 4: Short Answer (3 questions, 5 points each = 15 points)

Question 19. Explain the concept of "differencing" a time series. Why is it commonly used before fitting an ARMA model? Provide the mathematical definition of first-order differencing and give a sports example where differencing would be necessary.

Question 20. A bettor observes that an NFL team has gone 7-1 ATS (against the spread) in their last 8 games. Using time series reasoning, explain two different possible interpretations of this streak and describe how you would distinguish between them using the tools from Chapter 23.

Question 21. Describe how seasonal ARIMA (SARIMA) differs from standard ARIMA. Explain what the seasonal period represents in a sports context and give two examples of sports data where SARIMA would be more appropriate than ARIMA.

Section 5: Code Analysis (2 questions, 6 points each = 12 points)

Question 22. Examine the following Python code for fitting an ARIMA model:

import pandas as pd
from statsmodels.tsa.arima.model import ARIMA

def forecast_margin(series: pd.Series, order: tuple = (1, 1, 1)) -> float:
    model = ARIMA(series, order=order)
    fitted = model.fit()
    forecast = fitted.forecast(steps=1)
    return forecast.values[0]

# Usage
margins = pd.Series([3, 7, -2, 5, 10, 4, 8, -1, 6, 3, 12, 7])
next_margin = forecast_margin(margins)

(a) The code runs without error but has a significant methodological problem for a betting application. Identify the problem.

(b) Write corrected code that addresses this problem.

(c) What additional validation should be performed on the model before using its forecasts for betting?

import pandas as pd
from statsmodels.tsa.arima.model import ARIMA
from statsmodels.tsa.stattools import adfuller
from statsmodels.stats.diagnostic import acorr_ljungbox

def forecast_margin(series: pd.Series, order: tuple = (1, 0, 0)) -> dict:
    # Test stationarity to determine differencing
    adf_result = adfuller(series)
    d = 0 if adf_result[1] < 0.05 else 1

    actual_order = (order[0], d, order[2])
    model = ARIMA(series, order=actual_order)
    fitted = model.fit()

    # Check residual autocorrelation
    lb_test = acorr_ljungbox(fitted.resid, lags=[5], return_df=True)

    forecast = fitted.forecast(steps=1)
    conf_int = fitted.get_forecast(steps=1).conf_int()

    return {
        "forecast": forecast.values[0],
        "conf_int": conf_int.values[0],
        "aic": fitted.aic,
        "ljung_box_pvalue": lb_test["lb_pvalue"].values[0],
        "residuals_ok": lb_test["lb_pvalue"].values[0] > 0.05,
    }

Question 23. Examine the following changepoint detection code:

import numpy as np

def cusum_changepoint(data: list[float]) -> dict:
    """Detect a single changepoint using CUSUM."""
    n = len(data)
    mean = np.mean(data)
    cusum = np.cumsum(np.array(data) - mean)
    changepoint = np.argmax(np.abs(cusum))

    before = data[:changepoint + 1]
    after = data[changepoint + 1:]

    return {
        "changepoint_index": changepoint,
        "mean_before": np.mean(before),
        "mean_after": np.mean(after),
        "cusum_values": cusum.tolist(),
    }

(a) Trace through the execution with data = [10, 12, 8, 11, 3, 2, 5, 1]. Show the CUSUM values and identify the changepoint.

(b) What happens if data has no actual changepoint (all values drawn from the same distribution)? Will the function still report one?

(c) Suggest an improvement to handle the case where no changepoint exists.

def cusum_with_significance(data, n_permutations=1000, alpha=0.05):
    result = cusum_changepoint(data)
    observed_max = max(abs(c) for c in result["cusum_values"])

    null_maxima = []
    for _ in range(n_permutations):
        shuffled = np.random.permutation(data)
        cusum = np.cumsum(shuffled - np.mean(shuffled))
        null_maxima.append(np.max(np.abs(cusum)))

    p_value = np.mean(np.array(null_maxima) >= observed_max)
    result["p_value"] = p_value
    result["is_significant"] = p_value < alpha
    return result

Section 6: Applied Problems (2 questions, 8 points each = 16 points)

Question 24. An NFL team's game-by-game point differentials for the first 12 weeks of the season are:

[14, 7, -3, 10, 3, -7, -14, -10, -3, -7, -17, -6]

(a) (2 points) Compute the sample mean and standard deviation of the full 12-game series.

(b) (2 points) The team's pre-season projected point differential was +2.5 per game. Using a one-sample t-test, determine whether the observed performance is significantly different from the projection at the 5% level.

(c) (2 points) Apply CUSUM analysis to identify if and when a performance changepoint occurred. Report the CUSUM values and the estimated changepoint.

(d) (2 points) A sportsbook currently has the team as a 3.5-point underdog in Week 13. Based on your time series analysis, do you see value in betting the team or betting against them? Justify your answer using both the overall mean and the post-changepoint mean (if one exists).

Question 25. You are building a time series model to predict NBA team scoring for the purpose of betting game totals (over/under). You have three seasons of game-by-game data for all 30 teams.

(a) (2 points) Would you build one model per team or one model for the entire league? Justify your choice, considering sample size, parameter stability, and computational cost.

(b) (2 points) You fit an ARIMA(1,0,1) model to a single team's points-per-game series. The fitted AR(1) coefficient is 0.35 and the MA(1) coefficient is -0.20. Interpret these coefficients in the context of NBA scoring.

(c) (2 points) Your model's one-step-ahead RMSE on a held-out test set is 10.2 points. The sportsbook's game total has a typical margin of 2-3 points between the over and under lines. Is your model precise enough to be useful for betting? Explain.

(d) (2 points) Propose a strategy that combines your time series model with the sportsbook's game total to identify betting opportunities. Be specific about the decision rule and any thresholds you would use.

Scoring Summary

Grade Thresholds