ISS 2016 Statistics Paper-2 Solution: Question 17 (Writing the R Matrix for Linear Restrictions)

Continuing our question-by-question walk through the ISS Statistics Paper-2 archive, today's stop is 2016, Question 17. It's a short one on paper, but it hides a classic trap around how many parameters actually belong in the model — exactly the kind of trap that costs marks in the real exam if you rush past it.

Quick Summary

• Topic:Writing linear restrictions in the matrix form Rβ = r

• Question Reference:ISS 2016, Statistics Paper-2, Question 17

• Correct Answer:Option (a)

The Question, As Asked

What is R in linear hypothesis H0 : Rβ = r with two restrictions α0 = 2α1 and 3α1 = 4α3 in three variable linear model?

• (a) Row 1: [1, −2, 0, 0]; Row 2: [0, 3, 0, −4]

• (b) Row 1: [0, −2, 0, 0]; Row 2: [1, 3, 0, −4]

• (c) Row 1: [3, 0, 1, 0]; Row 2: [0, −2, −4, 0]

• (d) Row 1: [1, 0, −2, 0]; Row 2: [3, 0, −4, 0]

Step 1: What Rβ = r Is Actually Saying

In the general linear hypothesis testing framework, we don't test restrictions on the parameters one at a time using plain algebra. Instead, we collect every restriction into a single matrix equation:

Rβ = r

Here β is the full column vector of every parameter in the model, R is a matrix whose rows encode the restrictions, and r is a column vector of constants (often all zeros, when the restrictions only relate parameters to each other rather than to a fixed number). If there are two independent restrictions, R must have exactly two rows — one row per restriction, and one column for every parameter in β.

Step 2: Count the Parameters — The First Trap

The question says "three variable linear model." A three-variable linear model means there are three explanatory variables, say x1, x2, x3, being used to explain y. Written out, the model looks like:

y = α0 + α1x1 + α2x2 + α3x3 + ε

Notice that this gives four parameters in total — the intercept α0 plus the three slope coefficients α1, α2, α3. It is very tempting to read "three variable model" and assume there are only three parameters (dropping the intercept), and that single misreading is exactly what makes options (b) and (d) look plausible if you don't slow down here. Since the restrictions given to us explicitly mention α0, α1, and α3, we know for certain the parameter vector is:

β = (α0, α1, α2, α3)T

That's a 4×1 column vector, so R must have exactly 4 columns, in exactly this order.

Step 3: Convert Each Restriction Into an Equation Equal to Zero

Every row of R comes from taking one restriction and rearranging it so that everything sits on one side, equal to zero. This is the same habit you'd use to convert any word constraint into a standard linear equation — move every term to the left, leave zero on the right.

Restriction 1:α0 = 2α1

Move 2α1 to the left side:

α0 − 2α1 = 0

Now write this as a dot product with β = (α0, α1, α2, α3). The coefficient of α0 is 1, the coefficient of α1 is −2, and since α2 and α3 don't appear in this restriction at all, their coefficients are 0. So the first row of R is:

[1, −2, 0, 0]

Restriction 2:3α1 = 4α3

Move 4α3 to the left side:

3α1 − 4α3 = 0

Here the coefficient of α0 is 0 (it doesn't appear), the coefficient of α1 is 3, the coefficient of α2 is 0 (it doesn't appear either), and the coefficient of α3 is −4. So the second row of R is:

[0, 3, 0, −4]

Step 4: Stack the Rows to Get R

Putting both rows together, in the same column order (α0, α1, α2, α3) throughout, gives:

• Row 1: [1, −2, 0, 0]

• Row 2: [0, 3, 0, −4]

This is a 2×4 matrix — two rows because there are two restrictions, four columns because there are four parameters. And with r = (0, 0)T, you can check that Rβ = r reproduces exactly α0 − 2α1 = 0 and 3α1 − 4α3 = 0, which are just our two original restrictions rewritten. This matches option (a) exactly.

Pro Tip: The fastest way to eliminate wrong options here is to check the position of the zero columns before checking any signs. Restriction 1 never mentions α2 or α3, so the 3rd and 4th entries of row 1 must both be zero — that alone rules out option (c), whose first row has a 1 sitting in the α2 column. Similarly, restriction 2 never mentions α0 or α2, so the 1st and 3rd entries of row 2 must be zero — that alone rules out option (b), which puts a 1 in the α0 column of row 2. Building this habit of checking "which columns must be zero" before touching the numbers is exactly the kind of shortcut that gets drilled into muscle memory when this topic is taught properly, and it turns a question that looks like it needs careful matrix algebra into something you can solve by inspection in under a minute.

Step 5: Confirm Against the Official Answer Key

The official ISS 2016 Statistics Paper-2 answer key (Series A) lists the answer to Question 17 as option (a). This matches our derivation exactly, so there's no discrepancy to flag here — both the independent working and the official key agree that each restriction becomes exactly one row of R, built by moving every term to one side and reading off the coefficients in a fixed column order.

Why This Question Matters

The general linear hypothesis Rβ = r is the backbone of hypothesis testing in linear and regression models — it's what lets you test anything from "is a single coefficient zero" to "are several coefficients related to each other" using one unified F-test formula. If you can't reliably translate a plain-English restriction into a row of R, you can't set up that F-test at all, no matter how well you know the rest of the theory. Questions like this one are really testing a small, mechanical skill —correctly identifying every parameter in the model before you start writing rows — and that skill is exactly what trips up otherwise well-prepared candidates under exam pressure.

Frequently Asked Questions

Why does a "three variable" linear model have four parameters, not three?

"Three variable" refers to the number of explanatory (independent) variables in the model, not the number of parameters. Every linear regression model also carries an intercept term in addition to the slope coefficient for each explanatory variable, so three explanatory variables give you one intercept plus three slopes, for a total of four parameters.

What does r represent in Rβ = r, and why is it zero here?

r is a column vector holding the constant that each restriction equals once every parameter term has been moved to one side. Both restrictions in this question relate parameters to each other (α0 to α1, and α1 to α3) rather than fixing any parameter to a nonzero number, so both entries of r come out to zero.

Does the order of the columns in R matter?

Yes, absolutely. Each column of R must correspond to the same parameter, in the same order, as the corresponding entry of β. If you accidentally swap the column order partway through, you get a matrix that looks similar to the correct one but tests the wrong restrictions entirely, which is exactly the trap built into the incorrect options here.

How many rows will R always have?

Exactly as many rows as there are independent restrictions being tested simultaneously. Here there are two restrictions, so R has two rows; if the question had given three restrictions, R would need three rows, regardless of how many parameters the model has.

Can this same method be used for more complicated restrictions, like ones involving sums of several parameters?

Yes. The method never changes: move every parameter term of a restriction to one side of the equation, leave a constant on the other side, and read off the coefficient of every parameter (using zero for any parameter that doesn't appear) to get one row of R and one entry of r. This works no matter how many parameters are involved or how many appear in a single restriction.

Is this topic likely to reappear in later ISS papers?

Very likely. The general linear hypothesis is a recurring theme across ISS Statistics papers because it underlies so much of testing theory in linear models, so it's worth being completely comfortable with translating restrictions into R and r before moving on to the F-test itself.

If anything here felt unclear, or you solved it a different way, drop a comment below — working through the disagreement is often where the real learning happens. And if you know a fellow ISS aspirant who's grinding through the same paper, sharing this might save them a few minutes of confusion on exactly this question.