Why Consider a Regression Through the Origin?
In an ordinary simple linear regression model, an intercept allows the conditional mean of the response to be nonzero when the predictor is zero. Sometimes the subject matter suggests that the mean response should instead be zero at that point. A regression through the origin imposes that restriction by removing the intercept. This can give a useful model when the restriction is supported by the process being studied, but it is not a harmless simplification: it changes what the model says about the response.
The central question is not whether a scatterplot looks roughly like a line. It is whether a zero predictor should imply a zero conditional mean for the response, under the particular definitions and measurement units in use. The model is aligned with the SOA VEE Mathematical Statistics syllabus.
The Restriction at Zero
Substituting \(x=0\) into the conditional mean gives \(E[Y\mid X=0]=0\). Thus, the model says that the average response among cases with predictor value zero is zero. This is a statement about a conditional average, not a claim that every response is zero at \(X=0\). An individual response still includes its error term.
The restriction depends on what \(X=0\) means in context. If zero means “no production,” the restriction may be plausible for a response defined as production-related material cost. If zero means “age zero,” it is unlikely to be plausible for annual medical cost. A change in the definition or centering of \(X\) changes the meaning of the restriction.
Worked Example: Testing a Proportional Force Model
In a hypothetical spring experiment, \(X\) is extension in centimetres and \(Y\) is measured force in newtons. Over a small range of extensions, a physical model proposes \(Y=12X+\varepsilon\), with \(E[\varepsilon\mid X=x]=0\). Determine the model’s mean force at extensions of 0, 0.5, and 2 centimetres, and explain what it says at zero extension.
Method. Apply the conditional-mean equation \(E[Y\mid X=x]=\beta_1x\), using \(\beta_1=12\) newtons per centimetre.
At \(x=0\): $$ E[Y\mid X=0]=12(0)=0\text{ N}. $$ At \(x=0.5\): $$ E[Y\mid X=0.5]=12(0.5)=6\text{ N}. $$ At \(x=2\): $$ E[Y\mid X=2]=12(2)=24\text{ N}. $$
The restriction is that the mean measured force at zero extension is zero. It does not say that every measurement equals its mean: for example, an observed reading of 0.4 N at zero extension could arise from measurement error. The proposed proportional relationship is defensible only over a range where the spring behaves approximately linearly and the zero-extension calibration is appropriate.
Check. Each predicted mean is 12 times the extension, and the prediction at zero extension is zero as required.
Assumptions and What They Do
The equation \(Y=\beta_1X+\varepsilon\) defines the model’s form, but it does not by itself ensure that the model describes the conditional mean. The key assumption for that interpretation is \(E[\varepsilon\mid X=x]=0\) for predictor values in the model’s range. If the errors have a nonzero conditional mean, then \(E[Y\mid X=x]=\beta_1x+E[\varepsilon\mid X=x]\), and the claimed conditional mean need not hold.
For the classical simple linear regression framework, additional assumptions commonly used for inference are that errors have constant conditional variance, are independent across observations, and—when normal-theory inference is needed—are normally distributed conditional on the predictor values. These assumptions concern the errors, not the zero-intercept restriction itself. They should be justified by the data collection process and application; for example, responses that are counts or strongly bounded may not be well represented by constant-variance normal errors.
Define what one observation represents, what \(X\) measures, and what \(Y\) measures. State the units and the meaning of \(X=0\).
Ask whether a zero predictor should imply a zero conditional average response. Consider whether there is a baseline, startup amount, or measurement offset.
Consider whether errors have conditional mean zero and whether constant variance, independence, and any normality assumption needed for inference are reasonable.
If zero is outside the observed predictor range, the restriction may be difficult to assess from the observations. A compelling interpretation at zero still requires a defensible model over the relevant range.
The conditional-mean formula follows directly from the model and the zero conditional mean assumption. For a fixed predictor value \(x\), take the conditional expectation of both sides:
The second line uses the fact that conditioning on \(X=x\) makes \(X\) equal to \(x\), while \(\beta_1\) is a constant. The third line uses the zero conditional mean assumption. In particular, setting \(x=0\) yields \(E[Y\mid X=0]=0\). If \(E[\varepsilon\mid X=x]=0\) does not hold, this derivation stops at \(\beta_1x+E[\varepsilon\mid X=x]\).
Worked Example: Batch Cost With and Without a Setup Charge
A hypothetical workshop models \(Y\), the variable materials cost in dollars for a batch, using \(X\), the number of units in the batch. Suppose \(E[\varepsilon\mid X=x]=0\) and the proposed model is \(Y=7.50X+\varepsilon\). Calculate the conditional mean costs for batches of 0, 40, and 120 units. Then determine whether the same through-origin model is appropriate if the recorded response includes a fixed setup charge of $180 per batch.
Method. First apply \(E[Y\mid X=x]=7.50x\) to the response defined as variable materials cost only.
For zero units, \(7.50(0)=\$0.00\). For 40 units: $$ 7.50(40)=\$300.00. $$ For 120 units: $$ 7.50(120)=\$900.00. $$ These are conditional mean costs under the stated model; they need not be the exact cost of each individual batch.
If the response includes a fixed setup charge, its conditional mean would have a baseline of $180 even when \(X=0\). For example, with the same variable cost per unit, that mean relationship would be \(180+7.50x\), so at zero units it is $180 rather than zero. A through-origin model for total cost would impose the wrong zero-predictor restriction in that version of the problem. It may remain suitable for the variable-cost response, provided the other model assumptions are reasonable.
Check. The variable-cost predictions are zero, $300.00, and $900.00. Adding the setup charge would make the mean total cost at zero units $180, contradicting the through-origin restriction.
When the Restriction Is Defensible
A through-origin model can be sensible when the response is defined so that a zero predictor has a zero expected response, and when a proportional mean relationship is plausible over the range being analyzed. Examples might include the force generated by a suitably calibrated spring at zero extension, or the quantity of a material used in making zero units when there is no fixed setup use. These are hypothetical applications, not claims that the assumptions automatically hold in every such setting.
The restriction is not justified merely because the response and predictor are both measured numerically, because the relationship appears roughly proportional, or because a regression calculation is simpler without an intercept. An omitted baseline can produce systematic errors in the conditional mean. Conversely, if zero is far outside the observed range, the data may offer little direct evidence about the mean at zero. That limitation does not turn the restriction into a fact; it makes subject-matter reasoning especially important.
Worked Example: Annual Cost and Age as the Predictor
A hypothetical analyst considers \(Y\), annual health cost in dollars, and \(X\), age in years, for adults aged 30 to 60. A proposed through-origin model has \(\beta_1=95\) dollars per year and zero-mean errors conditional on age. Calculate the predicted conditional means at ages 30 and 60, state the model’s implication at age zero, and assess whether that implication is defensible for annual health cost.
Method. Use the model’s conditional mean \(95x\), then interpret its required value at \(x=0\).
At age 30: $$ 95(30)=\$2{,}850. $$ At age 60: $$ 95(60)=\$5{,}700. $$ At age zero: $$ 95(0)=\$0. $$
The model requires a zero mean annual health cost at age zero. That is generally not a defensible restriction for a response defined as all annual health costs. Also, the observations cover ages 30 to 60, so there are no observed cases at age zero to directly assess that implication. The computations are arithmetic consequences of the proposed model, not evidence that it is appropriate.
Check. Both calculated means are 95 dollars times age, and the formula forces the mean to zero at age zero. The lack of observations near zero does not remove that model restriction.
Common Mistake: Confusing the Mean With Every Observation
A frequent error is to interpret “through the origin” as saying that every observed point must lie on the line \(Y=\beta_1X\). The model includes an error term, so observations may lie above or below that line. For example, suppose \(\beta_1=12\) and an observation has \(X=0\), with an observed error of 0.4 N. The model gives \(Y=12(0)+0.4=0.4\) N for that observation. The conditional mean at \(X=0\) is still zero if the conditional mean of the error is zero.
The incorrect calculation is to set \(Y=0\) for every zero-predictor observation solely because the regression is through the origin. The correction is to distinguish the prediction of the conditional mean, \(E[Y\mid X=0]=0\), from a particular response, \(Y=\varepsilon\), which can differ from zero. The model’s central restriction is about the mean, not each realized response.
Check Your Understanding
For each problem, identify the model’s zero-point implication before deciding whether the restriction is plausible.
- A hypothetical calibrated sensor records response \(Y=4.2X+\varepsilon\), where \(X\) is input in volts and \(E[\varepsilon\mid X=x]=0\). Find the conditional mean responses at \(X=0\) and \(X=3\). (Answer: 0 and 12.6 response units.)
- A model for total shipment cost is \(Y=6X+\varepsilon\), where \(X\) is the number of shipped packages and the error has conditional mean zero. A separate fixed dispatch fee of $25 is included in \(Y\). What does the through-origin model predict at zero packages, and why is that restriction inconsistent with the fee? (Answer: It predicts a conditional mean of $0, but the stated fee makes the total cost at zero packages $25.)
- For a hypothetical materials-only production response, \(E[Y\mid X=x]=8x\), where \(X\) is units produced and \(Y\) is dollars of materials cost. Calculate the means at \(X=0, 15,\) and \(50\), and say whether the means alone guarantee every batch cost is exactly on the line. (Answer: $0, $120, and $400; no, individual responses may differ from their conditional means.)
- A researcher records annual income \(Y\) against years of work experience \(X\), but all observed workers have between 5 and 30 years of experience. What conditional-mean restriction does a through-origin model impose at \(X=0\), and what limitation follows from the observed range? (Answer: It imposes \(E[Y\mid X=0]=0\); there are no observations at zero experience to directly assess that restriction.)
- Suppose \(Y=\beta_1X+\varepsilon\), \(\beta_1=3\), and \(E[\varepsilon\mid X=x]=2\) for all \(x\). What is \(E[Y\mid X=x]\), and does the model imply a zero conditional mean at \(x=0\)? (Answer: \(E[Y\mid X=x]=3x+2\); no, at \(x=0\) the conditional mean is 2.)