## 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}.

## Statistical vs. Economic Significance

Last week, I learned how to distinguish the **statistical significance** and **economic significance **while doing the **regression analysis** in my econometrics class.

**-Statistical Significance:** We will look at the **t-tests** or **p-values** to determine whether or not to reject the null hypothesis (which says that the parameter is equal to 0) at a certain level of significance.

+ Statistical significance can be driven from a large estimate or a small standard error (which may result from a large sample size, meaning there are more variance in x variables)

+ A lack of statistical significance may be driven from small sample size or multicollinearity (meaning that there are correlations between x variables)

**-Economic significance**: we will look at the **magnitude** and the **sign** of the estimated coefficient. If the number turns out to be so small, that x variable does not really affect y.

### In short, **an coefficient is **

** statistically significant when it is quite precisely estimated,**

### and

### economically significant when it is important.

Here is one example from my class notes:

*#car*s: numbers of cars

*inctotal*: total annual income (in dollars)

*familysize*: the size of family (number of people)

*age*: measured in years

In the example above, all the parameters are **statistically significant** (the **t-ratios** are 8.81, 9.91, 2.26, 3.36, which is reasonable for us the *reject the null hypothesis* in a significance test).

However, the** magnitude** of “age” (measured in years) is really **small **(.0046911), which means a year increase in age will increase the number of cars owned by .0046911, on average, all else equal. In other words, all else equal, on average, in order to the number of cars owned to increase by 1, age must increase by over 200 years (=1/.0046911), which does not sound realistic!!

In conclusion, **even though the estimated coefficient of “age” is statistically significant****, it is NOT economically significant****.**

**Note: **The **magnitude **of “inctotal” is a small number too (.0000691), but it makes some sense. That means all else equal, a dollar increase in total income will increase the number of cars owed by .0000691, on average. That sounds possible, since *if you had an extra dollar per year, you would not think of buying another car*!! Well, it still does not make good sense to you, let’s put it this way: all else equal, on average, in order for a person to own another car, his/her annual income needs to increase by about $14,471 (=1/.0000691). Sounds reasonable, right? Therefore, **the estimated coefficient of “inctotal” is both statistically and economically significant**.

We need to watch out for the **units** as well. A small number doesn’t necessarily imply an economic insignificance.