I am trying to create a model for inflation for trading purposes. In his book The Market: Practice and Policy S. Nickell presents a model that relates unemployment, inflation and trade deficit. His final formula is

$$[ \alpha_1 + \delta_1 \alpha_{12}]u + \alpha_2 \Delta^2 p + \alpha_{12} \delta_2 td = [\alpha_1 + \delta_1 \alpha_{12} ] \hat{u}$$

After he fits this model to data, he finds the coefficients as,

$$0.091 \log u + 0.05 u + 1.07 \Delta^2 p + 1.25 td = 0.091 \log \hat{u} + 0.054 \hat{u} - 1.27 \Delta u$$

where $\Delta^2 p$ the rate of change of the price level (ie inflation), $u$ unemployment rate, $td$ trade deficit as proportion of potential output, $\hat{u}$ is natural rate of unemployment. The full derivation can be found at the link below

Nickell

I am trying to fit his formula to the data for UK, but cannot figure out how to get $\hat{u}$. Nickell seems to indicate this comes from a seperate calculation, I guess a first-pass on data would calculate $\hat{u}$, then with this new column in hand, I could fit all of the variables shown above. How to compute that first pass? Nickell says $\hat{u}$ can be defined "as that unemployment rate which is consistent with constant inflation and balanced trade" i.e. $\Delta^2p = 0$ and $td=0$. I am not sure what to do with this information: if I set $\Delta^2p = 0$, $td=0$ in the first formula above, I have

$$[ \alpha_1 + \delta_1 \alpha_{12}]u = [\alpha_1 + \delta_1 \alpha_{12} ] \hat{u}$$

which make no sense. What should my approach be for this computation? Any help would be greatly appreciated. Data for UK is below.

Data

I'm ot sure if it's the answer you're looking for but one commonly used method in practice is to simply take a long term average of the unemployment rate. The long term in this context means a period which covers exactly a full business cycle (either peak to peak or trough to trough).

FYI. US business cycle dates can be found here (http://www.nber.org/cycles.html). A google search will return similar for the UK.

From Wooldridge's Introductory Econometrics, 4th edition, pg. 390,

$$\Delta inf_t = \beta_0 + \beta_1 unem_t + e_t$$

Once we know the coefficients, natural rate can be computed as

$$\mu_0 = \beta_0 / (-\beta_1)$$

I worked out the code below,

import pandas as pd
df['dinf'] = df.inf.diff()
import statsmodels.formula.api as smf
results = smf.ols('dinf ~ unem', data=df).fit()
print results.summary()
print 'natural unemployment', results.params.Intercept / -results.params.unem


gives

5.463


meaning a natural rate of unemployment of about %5.5 - this is a reasonable number.

Data

 year       unem        inf
1948        3.8        8.1
1949        5.9       -1.2
1950        5.3        1.3
1951        3.3        7.9
1952          3        1.9
1953        2.9         .8
1954        5.5         .7
1955        4.4        -.4
1956        4.1        1.5
1957        4.3        3.3
1958        6.8        2.8
1959        5.5         .7
1960        5.5        1.7
1961        6.7          1
1962        5.5          1
1963        5.7        1.3
1964        5.2        1.3
1965        4.5        1.6
1966        3.8        2.9
1967        3.8        3.1
1968        3.6        4.2
1969        3.5        5.5
1970        4.9        5.7
1971        5.9        4.4
1972        5.6        3.2
1973        4.9        6.2
1974        5.6         11
1975        8.5        9.1
1976        7.7        5.8
1977        7.1        6.5
1978        6.1        7.6
1979        5.8       11.3
1980        7.1       13.5
1981        7.6       10.3
1982        9.7        6.2
1983        9.6        3.2
1984        7.5        4.3
1985        7.2        3.6
1986          7        1.9
1987        6.2        3.6
1988        5.5        4.1
1989        5.3        4.8
1990        5.6        5.4
1991        6.8        4.2
1992        7.5          3
1993        6.9          3
1994        6.1        2.6
1995        5.6        2.8
1996        5.4          3
1997        4.9        2.3
1998        4.5        1.6
1999        4.2        2.2
2000          4        3.4
2001        4.8        2.8
2002        5.8        1.6
2003          6        2.3