## Single Linear Combinations of Parameters

**Single Linear Combinations of Parameters means we are to test the linear relationship between two parameters in our multiple regression analysis. The simplest case can be**

**H _{0}:_{ } β_{1 }= β_{2}**

_{ }Or **H _{0}: β_{1 }= 10β_{2}**

Our hypothesis can be pretty much anything, as long as β_{1 }and β_{2} has linear relationship.

**Note** that we are to test whether or not **the effects of the two x variables** **on y** have **a linear relationship**, **NOT** the **linear relationship** between **the two x variables** **on each other** (that is the case of *perfect multicollinearity*).

For example, we are interested in testing

**H _{0}:_{ } β_{1 }= β_{2}**

**H _{1}:_{ } β_{1 }≠ β_{2}**

_{
}

**FIRST METHOD**

Set **θ = β _{1} – β_{2}**, then we will have

**H _{0}:_{ } θ = 0**

H_{1}:_{ } θ_{ }≠ 0

–**Set α** (if not given, assume it to be .05)

–**Find critical value**: df=n-k-1 (k is the number of x variables), then use the t-table to find critical value.

–**Calculate test statistic**:

*(That output above was an example from my class notes.)*

Therefore,

–**Decision**: to reject H_{0 }or not (by comparing t^{0}_{ }with the critical value)

–**Conclusion**:

If we **reject H _{0}**

_{ }(β

_{1 }= β

_{2}), we will conclude that

**β**

_{1}**is statistically different from β**.

_{2}at α levelIf we **fail to reject H _{0}**

_{ }(β

_{1 }= β

_{2}), we will conclude that

**β**.

_{1}is not statistically different from β_{2}at α level=========

Crazy enough, huh? There is another method that may look easier:

**SECOND METHOD**

-Set **θ = β _{1} – β_{2}**, then

**β**

_{1}= θ + β_{2}_{
}

_{ }

-Substitute **β _{1 }**in our original model by

**θ + β**

_{2}_{
}

y= β_{0 }+ **β _{1}**x

_{1 }+ β

_{2}x

_{2 }+ β

_{3}x

_{3 }+ u

y= β_{0 }+ **(θ + β _{2})**x

_{1}+ β

_{2}x

_{2 }+ β

_{3}x

_{3 }+ u =

**β**

_{0 }+ θ**x**

_{1}+ β_{2}(**x**

_{1}+x_{2})_{ }+**β**

_{3}x_{3}_{ }+ uNow our 3 variables in the model are **x _{1, }**

**x**

_{1}+x_{2, }**x**

_{3}_{
}

-Construct a new variable that is the sum of x_{1 }and x_{2 }(in STATA) by using the command

gen totx12 = x_{1 }+ x_{2}

_{“totx12” is just the name of the new variable.}

_{
}

_{ }

-Run the regression of y on x_{1}, totx12, and x_{3}

_{
}

-Test:

**H _{0}:_{ } θ = 0**

H_{1}:_{ } θ_{ }≠ 0

Now we can look at the t-ratio or p-value of the coefficient on x_{1} (**coefficient on x _{1 } is now θ**) then make our decision whether or not to reject H

_{0}.