## Wednesday, February 28, 2018

### Forecast performance of a quantity theory of labor

One of the dynamic information equilibrium model forecasts I've been tracking on the order of a year now to measure its performance is what I call the "N/L" or "NGDP/L" model [1] (specifically FRED GDP, i.e. nominal GDP, divided by FRED PAYEMS, i.e. total nonfarm payrolls). Revised GDP data came out today, so I thought it'd be a good time to check back in with the model [2]:

One way to think about this is as a measure of nominal productivity. We are coming out of the aftermath of the shock to the labor force following the great recession, so we can see a gradual increase back towards the long-run equilibrium.

If we use this dynamic equilibrium model instead of NGDP alone as the shocks, we can see in a history "seismograph" that this measure basically coincides with the inflation measures.

There's a good reason for this: this is effectively a model of Okun's law (as described here) if we identify the "abstract price" with the price level P:

$$P \equiv \frac{dNGDP}{dL} = k \; \frac{NGDP}{L}$$

which can be rearranged

\begin{align} L & = k \; \frac{NGDP}{P} \equiv RGDP\\ \frac{d}{dt} \log L & = \frac{d}{dt} \log RGDP \end{align}

to show changes in employment (and therefore unemployment) are directly related to changes in real GDP.

...

Footnotes

[1] Also, the "quantity theory of labor" per the title because the model implies log NGDP ~ k log L.

[2] Here is the complete model:

1. Time-series algebra is used for showing the precise relationship of RGDP to PAYEMS without needs of log-linearization, equilibrium and approximation.

1. %RGDP= %PAYEMS + %(RGDP/PAYEMS) + %PAYEMS * %(RGDP/PAYEMS)
2. %RGDP= %PAYEMS + %(RGDP/PAYEMS) if %PAYEMS * %(RGDP/PAYEMS) is (often) small.

The notations are defined as follows:
%RGDP = YoY % Change in RGDP
%PAYEMS = YoY % Change in PAYEMS
%(RGDP/PAYEMS) = YoY % Change in RGDP Per Payroll

In chart below,
Green line= %PAYEMS
Dot red line = %(RGDP/PAYEMS)
Blue line = %RGDP - %PAYEMS = Dot red line

https://fred.stlouisfed.org/graph/?g=iO9n

1. This is the just the identity

(d/dt) log X/Y
= (d/dt) (log X - log Y)
= (1/X) (dX/dt) - (1/Y) (dY/dt)

and would be true for any pair of variables X and Y (such as X = RGDP and Y = PAYEMS in your example). There is no model content unlike Okun's law described above which says

(d/dt) log X/Y ≈ 0 with X = RGDP and Y = PAYEMS

2. My question is: why we need to have approximate models for realistic economic data if we already have exact behavior expressions of component time-series?

In addition, approximate models would not guarantee to be true for future data, for example, time-series %RGDP is not equal to this expression "%PAYEMS + %(RGDP/PAYEMS)" if %PAYEMS * %(RGDP/PAYEMS) is not near to 0 during any past or future time periods.

My philosophical question about economic methodology is:
Are algebraic time-series expressions themselves behavior models to express time-variant relationships among economic parameters?

In genomics, we use gene expressions to represent the characteristics of gene relationships. In economics, IMO, we have to use algebraic expressions to represent the characteristics of time-variant relationships among economic data.

1. Your relationship between RGDP and PAYEMS is not an "exact behavior expressions of component time-series", but rather a mathematical identity that has no economic content.

You are conflating models with mathematical identities.

3. Jason,

Maybe I use a more specific example for your thoughts.
In general, we do not have a functional relationship between two economic time-series data, for example, y(t)=F(x(t)) since function F requires a unique y(t) by given the same x value at different time periods t's. Functional relationship is used inherently for time-invariant relationships.

But we can always use algebraic expressions to represent the time-variant relationships such as
(a) by ratio expression: y(t)= (y(t)/x(t)) * x(t),
(b) by spread expression y(t)= (y(t)-x(t)) + x(t)
(c) by any semantically-valid decomposition of y(t): assumed y(t)=z(t)*x(t), then %y(t) = %z(t) + %x(t) + %z(t)*%x(t), etc.

Aren't these algebraic expressions of time-series considered as behavior models?

1. y(t) = (y(t)/x(t))*x(t) is an identity, not a model

a = (a/b)*b = a

/ is the inverse of *

2. It depends on what economic time-series data that y(t) and x(t) refer to. Let me explain the differences of 4 mathematical identities in the following.

1. GDP(t)=(GDP(t)/PAYEMS(t))*PAYEMS(t)

2. GDP(t)=POTGDP(t)*(1-UnRate(t))
POTGDP(t) = GDP(t)*(1+Unemployment(t)/Employment(t))

3. GDP(t)= M(t)V(t)
V(t)= GDP(t)/M(t) M(t)=Supply of Money

4. GDP(t)= f(PAYEMS(t))+ error(t)
f is any regression function of time-series data.

Equations (1) and (2) above are accounting identities more than just mathematical identities. NIPA data such as GDP, PAYMENTS, etc. are structurally defined and measured. Based on Keynes' principle of effective demand, many equations "spending(effective demand)=income(effective supply)" are constructed in NIPA account. From this aspect, NIPA time-series data are "structural models" constructed from BEA/Fed by direct measurements of effective demands and supplies. Time-series PAYEMS, GDICOMP(Wages), Net Operating Surplus, etc. are effective supplies inside NIPA.

Equations (3) and (4) are not accounting identities. As you said, they are just math identities and external models needed by using composition approach. Equations (1) and (2) are based on internal structural models in NIPA by using decomposition approach.

Accounting version of Okun's Law should be something like this

%GDP = %POTGDP + %(1-UnRate)

This accounting version states that for every YoY k% decrease change in the employment rate(1-UnRate), a country's GDP will an additional k% lower than its potential GDP(POTGDP).

3. Equation (1) is an identity. We define what we mean by "/" and "*" by that equation which has nothing to do with the arguments of the binary operators *:A×A→A and /:A×A→A ... the set A could be anything.

a = a · b · b⁻¹

is always true because we define b · b⁻¹ ≡ e to be the identity element of the group where a · e ≡ a.

I will not discuss this further. If you'd like, you can read about the definition of a group:

https://en.wikipedia.org/wiki/Group_(mathematics)#Definition

4. Jason,
I am not talking math here. Based on principle of effective demand,
I can use the following equation to get the same result.

GDP(t)=(Wages(t)+COFC(t)+NetOperatingSurplus(t)+Tariffs(t))/PAYEMS(t))*PAYEMS(t))

1. Is

GDP(t) = Wages(t)+COFC(t)+NetOperatingSurplus(t)+Tariffs(t)

?

If yes, then it is just a contentless mathematical identity derivable from axioms about multiplication.

If no, then it is false.

2. Not sure what you are not understanding here. You seem capable of other math.

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