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Aligning the Pictures, Words and Numbers: Dynamic Models
© Edward E. Leamer
Turning Numbers into Knowledge
Page 1 of 4
Aligning the Pictures, Words and Numbers: Dynamic Models
Notes by Edward E. Leamer
Growth of Real GDP
It’s easy to align the pictures, words and numbers for GDP growth:
Growth Comes in Spurts
Picture
Growth of Real GDP
20
10
0
-10
-20
-20
-10
0
10
20
Growth of Real GDP(-1)
Words
Growth comes in spurts. A quarter of high growth tends to be followed by another quarter of high
growth. But there is a lot of noise in GDP growth and the tendency to have spurts of growth doesn’t help
forecast very much.
Numbers
Dependent Variable: G
Sample(adjusted): 1947:3 1998:2
Variable
C
G(-1)
R-squared
Adjusted R-squared
Coefficient
Std. Error
t-Statistic
Prob.
2.232
0.341
0.366
0.066
6.096
5.163
0.000
0.000
0.117
0.112
Mean dependent var
S.D. dependent var
3.401
4.365
Aligning the Pictures, Words and Numbers: Dynamic Models
© Edward E. Leamer
Turning Numbers into Knowledge
Page 2 of 4
Interest Rate on Three-Month Treasury Bill
It’s harder to align the numbers, pictures and words for interest rates
Here is my first equation to explain the interest rate on 3-month Treasury Bills.
Dependent Variable: R
Sample(adjusted): 1947:3 1998:2
Included observations: 204 after adjusting endpoints
Variable
C
R(-1)
R(-2)
R-squared
Adjusted R-squared
Coefficient
Std. Error
t-Statistic
Prob.
0.22
1.18
-0.22
0.10
0.07
0.07
2.30
17.16
-3.24
0.02
0.00
0.00
0.94
0.94
Mean dependent var
S.D. dependent var
5.03
2.99
These numbers are hard to explain. Why in the world is there a negative and statistically significant effect
of interest rates two quarters ago on this quarter’s interest rate? How can I find words that can possibly
explain that strange thing?
Let me tell you how. When you see changes in signs in dynamic equations, use changes of the variables.
Here is the suggested equation:
Dependent Variable: R
Sample(adjusted): 1947:3 1998:2
Included observations: 204 after adjusting endpoints
Variable
C
R(-1)
R(-1)-R(-2)
R-squared
Adjusted R-squared
Coefficient
Std. Error
t-Statistic
Prob.
0.22
0.96
0.22
0.10
0.02
0.07
2.30
57.37
3.24
0.02
0.00
0.00
0.94
0.94
Mean dependent var
S.D. dependent var
5.03
2.99
Verify that this equation is exactly the same in a mathematical sense to the one that precedes it. Verify
that 1.18 R(-1) - .22 R(-2) = .96 R(-1) + .22 ( R(-1) – R(-2)).
Now I have a story: Interest rates are set by the Federal Reserve Board. When Alan Greenspan decides
that he wants to increase interest rates, he doesn’t do it all at once. That might really jerk the economy
around. He does it slowly. Therefore when you see interest rates start to rise, it is telling us that interest
rates are on the move and will probably continue upward for a while.
Aligning the Pictures, Words and Numbers: Dynamic Models
© Edward E. Leamer
Turning Numbers into Knowledge
Page 3 of 4
Eviews advice: What is D(x)?
The Eviews way of writing a change is D(x) = x – x(-1). Check it out in this equation. Verify that it is
exactly the same as the one before.
Dependent Variable: R
Sample(adjusted): 1947:3 1998:2
Included observations: 204 after adjusting endpoints
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
R(-1)
D(R(-1))
0.22
0.96
0.22
0.10
0.02
0.07
2.30
57.37
3.24
0.02
0.00
0.00
R-squared
Adjusted R-squared
0.94
0.94
Mean dependent var
S.D. dependent var
5.03
2.99
Now you are ready to look at my picture, numbers and words for this variable:
Aligning the Pictures, Words and Numbers: Dynamic Models
© Edward E. Leamer
Turning Numbers into Knowledge
Page 4 of 4
Numbers
Dependent Variable: R
Sample(adjusted): 1947:4 1998:2
Included observations: 203 after adjusting endpoints
Variable
C
R(-1)
D(R(-1))
D(D(R(-1)))
R-squared
Adjusted R-squared
Coefficient
Std. Error
t-Statistic
Prob.
0.18
0.97
0.01
0.26
0.10
0.02
0.09
0.07
1.92
58.88
0.09
3.88
0.06
0.00
0.93
0.00
0.95
0.95
Mean dependent var
S.D. dependent var
5.05
2.98
Words
Take a good look at this regression output. You are probably wondering what is D(D(R(-1)))? You do
know what is D(R(-1)). That is the change in the interest rate. It’s positive when interest rates are rising,
and negative when interest rates are falling. It measures momentum. But that variable doesn’t matter; it
has a t-value of only 0.09. What matters is the change in the change, D(D(R(-1))) = D(R(-1)) – D(R(-2)).
That is an acceleration variable. Suppose that interest rates are going up. Like a baseball after it is hit.
When those rates start to slow their rate of ascent, then D(D(R(-1))) turns negative ; the most recent
increase is less than the previous increase. When that happens, there is a force to bring the interest rates
back to earth, just like a baseball.
Now the words: Greenspan has in mind a target level of interest rates. He isn’t going to jerk them up or
down to get to the target all at once. He wants a nice smooth path. When interest rates are going up, but
start to slow their rate of ascent, that’s a signal that there isn’t much more upward movement yet to come
and he may be getting ready to reverse direction. When rates are going down, and when they start to slow
their descent. That’s a sign that Greenspan has them as low as he dares, and he will probably start to move
them up again.
Interest Rates Have an Acceleration Effect
6
4
2
D(R)
Picture
0
-2
-4
-6
-4
-2
0
D(D(R(-1)))
2
4
6