| Course | BFO-7010 Business Forecasting |
|---|---|
| Module | Module 7 |
| Paper type | Module 7 written assignment: seasonal ARIMA modeling |
| Length | About 668 words, 5 pages |
| Format | APA 7 student paper |
| School | Thomas Edison State University |
| Program | MBA in Data Analytics |
| Updated | October 2026 |
Free sample paper for BFO 7010 Module 7
One Seasonal Difference and Two Parameters: An ARIMA Model for Propane Deliveries
Student Name
MBA in Data Analytics, Thomas Edison State University
BFO-7010: Business Forecasting
Mentor Name
Month Day, Year
One Seasonal Difference and Two Parameters: An ARIMA Model for Propane Deliveries
Step 1: Making the Series Stationary
An ARIMA model needs a stationary series, one whose mean and variance do not drift over time. Monthly gallons are far from stationary: the winter peak repeats every twelve months. Subtracting from each month the value twelve months before takes that pattern out. For 2020 to 2023 the seasonally differenced series has 48 values with a mean of 10,200 gallons, the average year-over-year growth, and no visible seasonal shape. Its time plot wanders around that mean without a lasting trend, so no further first difference is needed. Hyndman and Athanasopoulos (2018) warn against differencing more than necessary, since each extra difference adds noise and widens intervals.
Step 2: Reading the ACF and PACF
With 48 values, an autocorrelation beyond roughly plus or minus 0.29 stands out from noise at the 5 percent level. The lag-1 value of 0.28 sits at that edge and the partial autocorrelation at lag 2 is close to zero, about -0.01, the signature of a first-order autoregressive term: a month above its year-earlier level tends to be followed by another. The large negative spike at lag 12 is the classic sign of a seasonal moving-average term, often left behind by a seasonal difference. The lag-3 value is harder to explain and, with nothing in the business to account for it, is treated as a chance result to be checked in the residuals.
| Lag | Autocorrelation of the seasonal difference |
|---|---|
| 1 | 0.28 |
| 2 | 0.07 |
| 3 | 0.38 |
| 12 | -0.40 |
Step 3: Two Candidate Models
The first model uses only the AR(1) term and a constant on the seasonally differenced data. It leaves the lag-12 spike almost untouched, and its Ljung-Box statistic of 45.4 on 24 lags exceeds the 5 percent critical value of about 35.2 for 23 degrees of freedom, 24 lags less the one estimated coefficient (Ljung & Box, 1978), so its residuals still contain structure. Adding a seasonal moving-average term with a coefficient of -0.74 removes the lag-12 autocorrelation, lowers the residual standard deviation by a sixth and brings the Ljung-Box statistic to 26.1, well inside the critical value of about 33.9 for 22 degrees of freedom. The negative seasonal coefficient has a plain reading: when one January runs unusually high because of a cold snap, the model expects next January to give back most of that surprise rather than repeat it, which is how weather behaves.
| Measure | ARIMA(1,0,0)(0,1,0)12 with drift | ARIMA(1,0,0)(0,1,1)12 with drift |
|---|---|---|
| AR(1) coefficient | 0.29 | 0.28 |
| Seasonal MA(1) coefficient | none | -0.74 |
| Residual standard deviation | 39.9 | 33.0 |
| Residual autocorrelation, lag 12 | -0.34 | 0.02 |
| Ljung-Box Q, 24 lags | 45.4 | 26.1 |
| 2024 test RMSE | 35.9 | 26.8 |
| 2024 test MAE | 24.7 | 20.3 |
Step 4: Forecasting 2024
The second model's forecast for January 2024 was 652,000 gallons against an actual 618,200, and its 2024 total was 4.94 million gallons against 4.80 million. The one-step 95 percent interval is about plus or minus 65,000 gallons, widening over the year. Its test RMSE of 26.8 beats the seasonal naive benchmark (38.3), both Holt-Winters forms (30.7 and 31.4) and nearly matches decomposition with drift (26.9), though not the weather regression (24.5).
The forecast's main weakness is the drift. A constant of 13,000 gallons a year of growth, estimated from a period that ended with a cold winter, pushes the whole year about 3 percent high. Box et al. (2015) stress that the constant in a differenced model implies a permanent trend, so it should be included only when growth is expected to continue at that rate.
Step 5: Judging the Model
The seasonal ARIMA model passes the diagnostic checks and forecasts well without any outside data, which makes it a strong automatic option. It cannot, however, tell the dispatch manager why a month will be high. The pattern in the residuals of the Module 4 regression, an autocorrelation of 0.36, suggests the best model may combine both approaches: weather as a predictor with ARIMA errors. That is the dynamic regression of Module 8.
References
Box, G. E. P., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. (2015). Time series analysis: Forecasting and control (5th ed.). Wiley.
Hyndman, R. J., & Athanasopoulos, G. (2018). Forecasting: Principles and practice (2nd ed.). OTexts.
Ljung, G. M., & Box, G. E. P. (1978). On a measure of lack of fit in time series models. Biometrika, 65(2), 297-303. https://doi.org/10.1093/biomet/65.2.297
Reading the BFO 7010 Module 7 assignment instructions
ARIMA modeling is a likely subject for the written assignment in Module 7; BFO-7010 has six written assignments worth 40 percent in all, without a fixed module for each. Expect to test whether a series needs differencing, plot and read its ACF and PACF, propose and fit one or more ARIMA or seasonal ARIMA models, compare them by AICc or test accuracy, check residuals with a portmanteau test and forecast with intervals. Most versions ask for R's auto.arima result as a comparison with your own identification. The data here are a composite series built for this sample; yours will come from the textbook. Keep a record of each identification choice, since the written explanation is what the mentor grades.
How the BFO 7010 Module 7 example is put together
The paper follows the Box-Jenkins order in five numbered steps. Step 1 justifies one seasonal difference and explains why no more are taken. Step 2 tabulates the key autocorrelations, states the significance bound and matches each spike to a model term, flagging the one that has no explanation. Step 3 sets the two candidate models side by side and uses the Ljung-Box test to choose between them, then reads the seasonal coefficient in business terms. Step 4 reports the forecast, its interval and its error against every earlier model and names the drift as its weakness. Step 5 judges the model and points to the course's final method. Each claim about the model is backed by a number in the tables, from the significance bound to the Ljung-Box critical values.
Where the marks sit in the BFO 7010 Module 7 rubric
An ARIMA assignment is scored as part of the written work that makes up 40 percent of BFO-7010. Mentors look for justified differencing, ACF and PACF read correctly, a model whose terms follow from that reading, residual checks with a formal test and a forecast evaluated on unseen data. Reporting the coefficients without interpreting them earns little; explaining what a seasonal moving-average term means for the business earns more. Comparing the ARIMA results with the smoothing and regression models from earlier modules shows the evaluation of model validity that the first course objective names.
BFO 7010 Module 7 help from the desk
Points are commonly lost for differencing twice when once is enough, picking model orders from auto.arima without explaining them, ignoring a large seasonal spike in the residuals and quoting the Ljung-Box statistic without its degrees of freedom or critical value. Difference only as far as stationarity requires. Tie every term to a feature of the ACF or PACF. Run the residual test and state the result in a sentence. Decide whether the model needs a constant, and state the long-run growth that choice builds in. A free first sample of your own BFO-7010 Module 7 model can be requested from the desk. If the automatic routine picks a different model from yours, report both and explain the difference.
Write yours, or have the desk draft it
This paper is an original model document written by our desk, not a submitted student paper and not an official Thomas Edison State University document. Read it for the moves, then write your own to the instructions in your classroom. If you want one built to your exact prompt and rubric, the first custom sample is free and arrives in 24 to 48 hours.
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BFO 7010 Module 7 questions, answered
Where can I find a free BFO 7010 Module 7 sample paper?
The propane ARIMA assignment above seasonally differences the series, reads its ACF and PACF, fits two seasonal ARIMA models, applies the Ljung-Box test and forecasts the test year.
What does a spike at lag 12 in the ACF mean?
In monthly data after a seasonal difference, a negative lag-12 spike usually calls for a seasonal moving-average term.
How do I know if a series needs differencing?
A time plot that drifts or repeats a seasonal pattern, a slowly decaying ACF or a unit root test such as KPSS all suggest differencing.
What is the Ljung-Box test?
A test of whether a group of residual autocorrelations together differ from zero; a small p-value means the model has left structure behind.
What does ARIMA(1,0,0)(0,1,1)12 mean?
One autoregressive term, no ordinary differencing, one seasonal difference at lag 12 and one seasonal moving-average term.