Adaptive Expectations Model and Partial Adjustment Model

(This example is similar to Gujarati (2003) Example 17.10) We use the Hong Kong data, "hk_gdp_yearly" to run the adaptive expectations model and Partial adjustment model in this page.

First, create a Workfile like which includes the variables "consume", "gdp", and others.

We postulate the consumption is linearly related to permanent income: 

C_t = b₁ + b₂GDP^*_t + u_t

Since GDP*_t is unobservable, in order to generate the permanent income, a mechanism such as "adaptive expectations hypothesis is specified as:

 GDP^*_t - GDP^*_t-1 =  c  (GDP_t - GDP^*_t-1)

re-arrange as

GDP^*_t = c GDP_t - (1-c) GDP^*_t-1

Through substitution into the permanent income equation, we obtain 

C_t = b₁ + b₂[cGDPt + (1-c)GDP^*_t-1] + u_t

Consider lag one period and multiple it by (1-c)

(1-c) C_t-1 = (1-c) b₁ + (1-c) b₂[cGDP_t-1 + (1-c)GDP^*_t-2] + (1-c)u_t

Subtract from the previous equation, we get the autoregressive model as:

C_t = (1-c)b₁ + cb₂GDP_t + (1-c)C_t-1 + [u_t - (1-c)u_t]

==> C_t = a₁ + a₂GDP_t + a₃C_t-1 + v_t

Then, we can run this autoregressive model and obtain the coefficients for short run consumption function which will be use to deduce for coefficients of permanent consumption function. The equation specification is

The regression result is:

Estimated (C_t) = 1.9808 + 0.5310(GDP_t) + 0.0988(C_t-1)

This result shows that the short-run marginal propensity to consume (MPC) is 0.5310.

But if the increase in income is sustained, then eventually the MPC out of the permanent income will be

b₂ = a₂ / c = 0.5310 / (1-0.0988) = 0.5892

To test whether it has serial correlation in the autoregressive model, Durbin h test can be used:

Durbin h* =(1-DW/2)[n/1-(n*Var(b₃)]^(1/2)

h* = (1- 0.5*0.75659)*[29/(1-29(0.0693)²)]^1/2 = 3.60

Since h* > 1.96, thus at the 5% significant level we reject the H_o that there has the first-order autocorrelation in the autoregressive model.

In EVIEWS, we can use the pre-programmed Breusch-Godfrey (BG) test, also known as the Lagrange multiplier (LM) test to test whether there is any high-order autocorrelation within the model. Simply. by choosing "View", "Residual tests", " Serial Correlation LM Test", 

and set "Lags to include" is "1", then the result is:

As you can see, the computed n*R is 10.89 which is large and its probability is also close to 0,  we can conclude that there may exist a higher-order series correlations in the autoregressive model. (Remark: Since a high-order autocorrelation may exist in the model, we should use other methods to remove it , but it is out of our scope.)

For finding the desired long run coefficients, b₀, and b₁ of the permanent consumption function, the calculation are as following:

since (1- c)  = 0.0988 and c = 0.9012

b₀ = 1.9808 / (0.9012) = 2.1979

b₁ = 0.5310 / (0.9012) = 0.5892

The estimated permanent consumption equation is:  

C_t = 2.1979 + 0.5892 GPD^*_t

Partial Adjustment Model

Now suppose the consumption were the desired or long-run consumption is a linear function of the current or observed income as

C^*_t = d₁ + d₂GDP_t + u_t

Since C*t is not directly observed, and suppose the observed consumption is following the partial adjustment mechanism as

 C_t - C_t-1 =  q  (Ｃ^*_t - Ｃ_t-1)

Substitute the desired consumption equation into the partial adjustment, we have

C_t - C_t-1 =  q  [(d₁ + d₂GDP_t + u_t - Ｃ_t-1]

==> C_t = C_t-1 +  q d₁ +q d₂GDP_t + qu_t - qＣ_t-1

_==> C_t =  q d₁ +q d₂GDP_t +(1- q)Ｃ_t-1+ qu_t

_==> C_t =  a₁ + a₂GDP_t + a₃Ｃ_t-1+ V_t

In appearance, this autoregressive model is not different from the adaptive expectations model. Not to mention the estimation problem associated with the possible autocorrelation, the interpretation is quite different with the adaptive expectations model.

This partial adjustment model is the long-run, or equilibrium, consumption function, where d₂ is measuring the long-run MPC, while the estimated a₂ only gives the short-run MPC. Therefore, from the estimated a₂, the calculated the long-run MPC is 0.5892.

[d₂ = a₂/(1-a₃) = 0.531067/(1-0.098873) = 0.5892.

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