10.3 Applied Causal Forecasting: Regression, Correlation & Leading Indicators

Applied Causal Forecasting: Regression, Correlation & Leading Indicators

Time-series methods answer "what has this series been doing?" Causal (associative) models answer "what drives this series, and what are those drivers about to do?" That difference matters most exactly where time series fail — at turning points and over long horizons. Exam 2 tests both the arithmetic and the judgment.

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Simple Linear Regression — Fully Worked

The model fits a straight line $Y = a + bX$, where $Y$ is demand (the dependent variable) and $X$ is the driver (the independent variable), by minimizing the sum of squared vertical distances between the observations and the line — hence least squares.

$$b = \frac{n\sum XY - \sum X \sum Y}{n\sum X^2 - (\sum X)^2} \qquad a = \frac{\sum Y - b\sum X}{n} = \bar{Y} - b\bar{X}$$

Scenario: A supplier of commercial HVAC filters believes demand tracks regional non-residential construction permits issued two quarters earlier.

Step 1 — Slope.

$$b = \frac{6(1{,}605) - (42)(216)}{6(322) - (42)^2} = \frac{9{,}630 - 9{,}072}{1{,}932 - 1{,}764} = \frac{558}{168} = \mathbf{3.321}$$

Step 2 — Intercept.

$$\bar{X} = 42/6 = 7.0 \qquad \bar{Y} = 216/6 = 36.0$$ $$a = 36.0 - (3.321)(7.0) = 36.0 - 23.25 = \mathbf{12.75}$$

Step 3 — The forecasting equation.

$$\hat{Y} = 12.75 + 3.321X$$

Step 4 — Forecast. If permits two quarters ago were 12 (hundred):

$$\hat{Y} = 12.75 + 3.321(12) = 12.75 + 39.85 = \mathbf{52.6\ \text{thousand filters}}$$

Step 5 — Interpret the coefficients in business language.

• Slope $b = 3.321$: each additional hundred permits is associated with roughly 3,321 additional filters. This is the number a category manager uses to size a capacity commitment.
• Intercept $a = 12.75$: about 12,750 filters of demand exist independent of new construction — replacement and maintenance demand. Note the caution: the intercept is only meaningful if $X = 0$ falls within a plausible range of the data.

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Correlation and the Coefficient of Determination

$$r = \frac{n\sum XY - \sum X \sum Y}{\sqrt{\left[n\sum X^2 - (\sum X)^2\right]\left[n\sum Y^2 - (\sum Y)^2\right]}}$$

Continuing the example:

• Numerator = $558$ (computed above)
• $n\sum X^2 - (\sum X)^2 = 168$
• $n\sum Y^2 - (\sum Y)^2 = 6(8{,}086) - (216)^2 = 48{,}516 - 46{,}656 = 1{,}860$
• $r = \frac{558}{\sqrt{168 \times 1{,}860}} = \frac{558}{\sqrt{312{,}480}} = \frac{558}{559.0} = \mathbf{0.998}$
• $R^2 = (0.998)^2 = \mathbf{0.996}$

Reading these numbers:

Exam trap: a high $R^2$ describes fit to history, not forecast accuracy. Adding variables always raises $R^2$ mechanically, which is why adjusted $R^2$ and out-of-sample validation exist. A model with $R^2 = 0.98$ built on six observations and eight variables has fitted noise.

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Leading, Coincident and Lagging Indicators

A causal model is only useful if the driver is known before the demand it predicts.

Validating a candidate indicator — the practical checklist:

1. Is there a plausible causal mechanism? Without one, the correlation is probably spurious.
2. Is the lead time stable? A driver that leads demand by two quarters sometimes and five quarters at other times is not usable.
3. Is it published in time? An indicator with a three-month reporting delay and a two-month lead provides no actionable warning.
4. Is it revised? Many economic series are revised substantially after first release; a model built on first prints behaves differently in production.
5. Does it hold out of sample? Fit on one period and test on a period the model never saw.

For supply management specifically, the ISM Report On Business and its Purchasing Managers' Index are the classic leading indicators of industrial activity, with the new-orders sub-index leading production. Sector-specific leaders include housing starts and permits for building products, vehicle production schedules for automotive components, and rig counts for oilfield supply.

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Multiple Regression and Multicollinearity

Multiple regression extends the model to several drivers:

$$\hat{Y} = a + b_1X_1 + b_2X_2 + \cdots + b_kX_k$$

Example: filter demand as a function of construction permits, average regional temperature, and a regulatory air-quality index.

Multicollinearity occurs when the independent variables are strongly correlated with each other — for instance, including both GDP and industrial production. The model may still fit well overall, but the individual coefficients become unstable and can even take the wrong sign, so any interpretation of an individual driver's effect becomes unreliable. Remedies: drop one of the correlated variables, combine them into a single index, or collect more data.

Other standard cautions:

• Overfitting — too many variables relative to observations. A workable rule of thumb is at least ten observations per independent variable.
• Extrapolation beyond the observed range — the fitted relationship has no evidence outside the data. Forecasting demand at $X = 40$ from data spanning $X = 4$ to $X = 10$ is extrapolation, not prediction.
• Structural breaks — a new competitor, a regulation, or a pandemic changes the relationship itself. Historical coefficients then describe a world that no longer exists.
• Autocorrelation — successive residuals correlate in time-series regressions, which understates the true standard errors and makes the model look more reliable than it is.

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When to Use Causal Models in Supply Management

The strongest practical answer on Exam 2 is usually a blend: a statistical baseline for the short horizon, a causal overlay for the medium horizon, and structured judgment applied through the S&OP consensus process — with each adjustment documented so that forecast value added can be measured afterward.