# Chapter 6: Regression

Regression,
like correlation, does not determine causation. Its strength is that unlike
correlation, it measures the parameters of the association. That is, correlation
can show that disposable income and household consumption move together, but
regression measures the amount by which consumption will increase when disposable
income rises by a dollar. Because regression goes beyond correlation to a measurement
of the size of the affect of one variable on another, it is the favorite statistical
technique in empirical economics.

In the example
of regression just cited there is an implicit assumption about causation,
even though neither regression nor correlation can prove it. Economists assume
that changes in disposable income cause changes in consumption (although we
allow that the reverse is true as well, at least in the aggregate). In the
regression statistical procedure, is assumed that one variable is dependent
(consumption) and the other is independent (disposable income). This assumption
tells us about the researcher’s intuition, or the theory in use, but it cannot
be validated or invalidated with the regression procedure. To repeat, whatever
we think we know about causation must come from theories and histories and
other pieces of information besides regression.

6-1
The consumption function

Chart 10
in Chapter 5 plotted the pairs of values for consumption and disposable income
for each year, 1929-1996. In the economics literature, this is known as the
consumption function. Economists theorize that consumption is largely
determined by disposable income (after tax income). Algebraically, we can write
the consumption function in general functional notation:

C = f(Yd),

where C is consumption
and Yd is disposable income. The plot of c and dp1 in Chart 10
revealed that the relationship was linear, so we can convert the general functional
notation into a specific, linear functional form:

C = c0
+ c1 Yd.

In this form,
the consumption function is a straight line, with intercept c0
and slope c1. In economics, the intercept, c0, is called
autonomous consumption since it is independent of (autonomous from)
disposable income. The slope, c1, measures the rate of change in
consumption given a change in Yd. For example, if Yd
increases by \$1, then C changes by (c1 )*(\$1) = c1.

Let D stand for
the change in a variable, so D Yd is read as "the change in disposable
income." Then, if Yd changes by D Yd, the change in
C (D C) is c1 D Yd:

C = c0
+ c1 Yd,, and D C = c1D Yd,

so that c1 = D C /D Yd = the marginal
propensity to consume
= MPC

Autonomous consumption
and the marginal propensity to consume are the parametersof the linear
consumption function. Mathematically, they are the intercept and slope of
a line that describes the relationship between disposable income and consumption.
Regression analysis is an exercise in estimating their values, but before
we do regression, we have to take into account one more element of every regression
model.

The linear consumption
function, C = c0 + c1 Yd, is a deterministic
model
. It allows no room for variation away from the relationship. Once
Yd is known, C is completely determined (given the parameters c0
and c1). In fact, the consumption function describes a tendency,
not a mathematically fixed relationship. The relationship between consumption
and disposable income is probabilistic, or to say the same thing with
a 5 dollar word, it is a stochastic relationship. Stochastic relationships
are not fixed like deterministic relationships, there is always margin for
variation away from the general tendency. Therefore, if the consumption function
describes the deterministic relationship, we need to add a term to let the
actual behavior of consumption in the actual economy deviate from the value
predicted by disposable income:

C = c0
+ c1 Yd + e,

where e is a
random error term. On average, e is zero, so C = c0 + c1
Yd, but in any given year, C could be more than predicted (e >
0), or less than predicted (e < 0). Graphically, the inclusion of a random
error terms allows for the possibility that the scatter points of the consumption
function do not fall on a straight line.

With the regression
procedure in SPSS we can compute the values of the parameters c0
and c1.

1. Select
Statistics from the menu bar, choose Regression, then select Linear .
. .;
2. Highlight
c in the variable list, and click the arrow to put it into the Dependent
box;
3. Highlight
dp1 in the variable list, and click the arrow to put it into the Independent(s)
box;
4. Click OK.

The results are
in Table 6 where I have divided them into 3 parts. In each part, the
most important numbers are in bold. Part 1 has 1 number (Adj. R Square =.99964),
Part 2 has none, and Part 3 has several. One of the keys to using SPSS or any
statistical package, is to not become overwhelmed by the amount of output it
generates; the trick to that is to know what you can ignore, at least initially.
As you become more skilled, you will find uses for the things we are going to
ignore for now.

Table 6

Elements of Regression Output

 Part 1 Multiple R 0.99982 R Square 0.99965 Adj. R Square 0.99964 Standard Error 27.4352 Part 2 Analysis of Variance DF Sum of Squares Mean Square Regression 1 141122061.6 141122061.6 Residual 66 49677.6 752.7 F=187489.9 Signif F=0.000 Part 3 Variable B SE B Beta T Sig T DP1 0.918815 0.002122 0.99984 433.001 0.000 (Constant) -12.33177 4.243069 -2.906 0.0050

Part 1 provides
4 measures of goodness of fit. These are statistics that tell how well
the data fits the model. R Square and its adjustment, Adj. R Square, can be
interpreted as the percentage of the variation in the dependent variable that
is explained by the independent variable. In our model, C is the dependent variable
and Yd is independent, so movements in Yd explain nearly
all (>99%) of the movement in C. There is no threshold for the R squared or
adjusted R squared where they go from bad to good, but by any criteria, our
model explains nearly all the variation in C.

is an adjustment to R square (duh!) that takes into account the number of
independent variables. Since we only have one, Yd, the two are
close in value. The adjusted R squared of 0.99964 looks too good to be true
and it probably is; for various technical reasons, some statistical, some
economic, it makes the model look better than it is. (Two reasons: autocorrelation,
and nominal data.) It pays to be skeptical, even (especially) when things
look great.

Part 2 provides
a number of statistics that are grouped together under the subject of analysis
of variance
. Basically, Part 2 provides measures that break down the variation
in C and attribute the different parts to the deterministic part of the model
(c0 + c1 Yd ) and the stochastic part (e).
These are useful measures in more advanced routines, but they are unnecessary
at this point.

Part 3 is the
core of the output. Part 3 has the estimated values of c0 (-12.33)
and c1 (0.9188). These are in the column labeled B. The next column,
SE B, is the standard error of the estimates (0.002122 for c1,
and 4.243069 for c0.) These are measures of the precision of our
estimates of c0 and c1. The smaller the standard errors,
the more precise are our estimates. The column labeled Beta can be ignored,
but the following column, T, has important information. T is the value of
the t-statistic that is constructed to test the hypothesis that the "true"
values of c0 and c1 are zero. Let the unobserved true
values be symbolized with Greek letters, b0 and b1.
We want to test the hypotheses:

H0:
b0 = 0 versus H1: b0 &sup1; 0, and

H0: b1 = 0 versus H1: b1 &sup1;
0.

If we accept
the second null, then it means that disposable income has no affect on consumption.
Since this is one of the primary reasons for doing regression (i.e., to see
if disposable income affects consumption, and if so, how much), every statistical
package automatically turns out a t-statistic to test this hypothesis. The
formula for the t-statistic is:

t-value = (c1
- value in null hypothesis)/(standard error of the estimate) =

(B - 0)/(SE B) = (0.9188 - 0)/(0.002122) = 433.

The last column
of the SPSS printout in Part 3 is labeled Sig T. It is the probability of
the t-statistic, which is also the probability of getting the data in the
dataset when the null hypothesis is true (H0: b1 = 0).
Since the probability (to four decimal places) of getting a sample value,
c1 , that is 0.9188 with a standard error of 0.002122, is 0, we
should reject the null hypothesis.

6-2
Okun’s Law

Let’s try another
regression. (This section and the following borrow heavily from Blanchard, 1997.)
Economists have long known that increases in the rate of growth of GDP enables
more unemployed people to finds jobs. It was not until the 1960s and the work
of Arthur Okun that that this general relationship between unemployment and
GDP growth was estimated empirically. The question is simple and basic: If GDP
falls by 1%, how much does the unemployment rate change? In order to answer
this, we have to compute the change in the unemployment rate, ur - lag(ur).
We will call this variable Dur, where the Greek letter delta, D , indicates
change. We also have to compute the percent change in real GDP, which requires
two steps (you may already have done this). First, use the GDP deflator (gdpdef)
and GDP to compute real GDP. Second, compute the percentage change in real GDP,
and give it a name. Then you are ready to run the regression.

The estimated
equation is

Dur = 1.274321
- 0.361428(Percent change in real GDP).

The interpretation
of this relationship is that, on average, each 1% increase in the rate of
growth of real GDP, reduces the unemployment rate by 0.36 percent. You should
check the goodness of fit statistics, R square and adjusted R square, and
the t-statistics for the slope and the intercept. Follow the procedure outlined
for the consumption function.

The implications
of Okun’s Law are that output must grow by about 3.5% per year (1.27/0.36)
just to keep unemployment from rising. Why? The answer is that the labor force
grows about 1% a year (check this), so output has to grow at about the same
speed to provide enough new jobs. Second, labor productivity (output per hour
worked, prod1 in the dataset) grows at about 2.3 percent a year (check this)
so even if no new jobs are created, output goes up 2.3 percent. Put these
two forces together, and real GDP has to grow over 3 percent a year on average
just to keep the unemployment rate from going up. Because of this relationship,
many economists view "normal" economic growth as approximately 3-3.5%.

Okun’s Law has
also been used to try to measure the costs of unemployment to the national
economy. When unemployment holds constant (Dur = 0), real GDP grows about
3.5%. Now solve for the percent change in real GDP if unemployment rises by
1 percentage point (D ur = 1):

1 = 1.2743 -
0.36142(Percent change in real GDP)

Þ Percent change in real GDP = 0.75.

When GDP growth
falls from 3.5% to 0.75%, we lose about 2.75 percent of potential GDP. Given
that our GDP is roughly 8,000 billion in nominal terms, a loss of 2.75 percent
represents a loss of about \$220 billion (0.0275*8,000). In other words, each
1% increase of unemployment costs the US economy around \$220 billion in lost
output.

6-3
The Phillips curve, then and now

The Phillips curve
was one of the key economic discoveries of post-World War II macroeconomics.
Recall that the curve showed a regular relationship between inflation and unemployment.
This seemed to give policymakers a set of inflation-unemployment tradeoffs they
could choose. If inflation was too high, then use Keynesian policies to slowdown
the economy-- unemployment would rise, but the amount was predictable and did
not vary. If unemployment was too high, then do the opposite--inflation would
rise, but again it was predictable and invariant.

To see the Phillips
relationship that economists in the 1950s and 1960s worked with, we should
omit the data from the 1930s and World War II. In addition, since the relationship
broke down in the 1970s, we will work with data limited to 1948-1969. Algebraically,
the relationship can be expressed as

pt
= b0 + b1ut + et,

where pt
is inflation in year t, b0 is the intercept of the regression line,
b1 is the slope parameter which is expected to be negative, ut
is the unemployment rate in year t, and et is the random error
terms that measures deviations from the average relationship.

In the data set,
the unemployment rate is variable ur, and the inflation rate is a computed
variable that is the percentage change in the CPI. We calculated this in several
earlier exercises.

1. Select
Data from the menu bar, then Select Cases. . .;
2. Click the
button for Based on time or case range, then click Range;
3. In the
boxes type 1948 and 1969;
4. Click OK.

Now run the regression
using your inflation variable as the dependent variable and ur for the independent
variable. You should get

pt
= 6.917 - 0.987ut + et,

This is the relationship
that broke down during the 1970s. To see this, change Select Cases to the
years 1970 to 1996 and re-run the regression. Look at the R squared. Does
ut explain anything about inflation? Is the sign on ut
what you expected (i.e., is your estimate of b1 negative)? Is it
significantly different from zero? That is, do you accept or reject the null
hypothesis H0: b1 = 0?

Needless to say,
most economists were puzzled by this. As early as the mid-1970s it was apparent
that the Phillips relation no longer worked. What could have gone wrong? The
answer was waiting in the wings in the form of a earlier prediction made by
Milton Friedman. Friedman had argued that as soon as people changed their
expectations about inflation, the Phillips curve would breakdown. Friedman’s
point was that inflation partly depended on what people expected it to be.
If everyone thought it was going to be high, then workers would demand wage
increases, and businesses would expect higher costs, so they would raise their
prices. The net result would be inflation--in part because everyone expected
it and acted to protect themselves by raising their wage demands and their
prices.

Until the late
1960s, prices seemed to have no trend; they were about as likely to fall as
they were to rise. Consequently, it made sense to expect zero inflation since
that was close to the long term average. In the late 1960s and early 1970s,
this changed. Inflation was ratcheted up by a combination of events--the Vietnam
War, domestic spending for the War on Poverty, bad harvests in the early 1970s,
and, in 1973, the first oil crisis. Households and businesses began to expect
that inflation would not be zero, facts bore out the correctness of this view,
and the inflation rate rose. Friedman’s arguments led economists to the "expectations
augmented" Phillips curve, which is just the old Phillips curve with another
variable, expected inflation, on the right hand side:

pt
= b0 + pe + b1ut + et,

where pe
is the expected rate of inflation. The old Phillips curve is a variety of
this one in which pe is zero. Here, pe is expected to
be positive, so that for a given unemployment rate, inflation is higher by
that amount.

The obvious question
is whether or not this can be measured. That is, how do we know (measure)
the expected rate of inflation? Friedman’s answer was to point out that most
of us use the recent past to form our expectations about the future. For example,
will it be hot or cold today? When my kids ask me that in the morning, I always
tell them that it will be just like yesterday. (Of course, I could look it
up in the weather section of the morning paper, and sometimes I do if there
is reason to believe the weather might be changing. Looking it up--seeking
additional information--is the rational thing to do and conforms to the economic
idea of rational expectations. It is forward looking and incorporates
all readily available information that is not too costly to obtain. Friedman’s
idea--today is like yesterday--is called adaptive expectations.)

For the sake
of simplicity, we assume that our expectation of inflation today is that it
will be like last period’s rate. Algebraically,

pe
= pt-1,

where pt-1
is the inflation rate in year t-1 (i.e., last year if this is year t). Using
this notation, we can re-write the expectations augmented Phillips curve as

pt
= b0 + pt-1 + b1ut + et,

or, moving the
expected inflation term to the left:

pt
- pt-1 = b0 + b1ut + et,

which we can
easily estimate for 1970 to 1996. After selecting the years 1970 to 1996,
and computing a new variable pt - pt-1, re-run the regression.
You should get

pt
- pt-1 = 7.078 - 1.085ut + et.

Notice the similarity
to the regression for 1948 to 1969.

This regression
has many uses in policy making. For example, it implies that if unemployment
is too low, the left hand side will be positive and inflation will be accelerating
(pt > pt-1). Economists have a special fondness the
rate of unemployment that keeps inflation from rising. Note that this is not
the same thing as zero inflation. The unemployment rate that prevails when
pt - pt-1 equals 0 is known as (get ready!) the non-accelerating
inflation rate of unemployment
, or the NAIRU. A prettier but misleading
name for it is the natural rate of unemployment.

What is the natural
rate? Set the above equation equal to zero, and solve:

0 = 7.078 -
1.085ut,

or ut
= 6.5. Anything less, and inflation is supposed to increase; anything more
and it decreases. Unemployment is currently less than 5%, so you can guess
why the Federal Reserve and inflation hawks are nervous. Inflation should
be ratcheting up, but it is not. We don’t know why, and the debate rages on
among economists. It is clear, however, that the natural rate of unemployment,
or the NAIRU, changes over time. It seems to have fallen in the 1990s, but
no one can say how low. The data is not loud and clear enough for us to be
certain.

6-4
Sources

A full treatment
of the Phillips curve and Okun’s Law in a more theoretical framework can
be found in Blanchard, Olivier. Macroeconomics. New Jersey: Prentice
Hall. 1997.