# Link to the original text:http://tecdat.cn/?p=21506

When two states are adopted, the single transfer function PSTR model has two variables

The basis of our empirical approach includes assessing capital liquidity in n countries. The corresponding model is defined as follows:

Among them, it is the ratio of domestic investment to GDP observed in the ith country at time t, and sit is the ratio of domestic savings to GDP, α I is a single fixed effect. surplus ε It is assumed to be i.i.d. (0, σ two ε）。 Corbin (2001) uses this model in particular, which has two main disadvantages.

First, it assumes that the international mobility of capital among the n countries in the group is the same, i.e β i= β， ∀i=1，…，N。 Obviously, even if we only consider OECD countries, this assumption is unrealistic. As mentioned earlier, a number of factors have been identified that significantly affect capital flows: country size, age structure of the population, openness, etc. Therefore, it is assumed that β i= β This means that these factors do not affect capital flows. This assumption is obviously too strict.

Secondly, equation (1) shows that the saving retention coefficient is constant in the estimation period of the model. This assumption is also unrealistic, especially when we consider the macro panel with a long enough time dimension: it is clear that the capital liquidity of typical OECD countries is not the same in the 1960s and 1990s.

Since the mid-1970s, capital controls and barriers to cross-border capital flows in major OECD countries have been eliminated, and the FH coefficient shows a downward trend over time. In fact, Obstfeld and Rogoff (2000) found that the saving retention coefficient of OECD countries was 0.60 in 1990-1997, while FH emphasized that the saving retention coefficient of 16 OECD countries was 0.89 in 1960-74. Therefore, there is no reason to assume parameters β（ parameter β i) Time is constant.

Generally speaking, these two problems cannot be solved at the same time. For example, by assuming the FH parameter β I is randomly distributed to consider heterogeneous panel model 5. However, in such a random coefficient model, the liquidity of capital is assumed to be time invariant. In addition, in a simple random coefficient model (swamy, 1970), the parameter β I is assumed to be independent of the explanatory variable. In other words, assume that the FH coefficient has nothing to do with the ratio of domestic savings to GDP. Therefore, their variability is the result of other unspecified structural factors.

The way to solve these two problems is to introduce threshold effect into the linear panel model. In this case, the first solution is to use the simple panel threshold regression (PTR) model (Hansen, 1999), as suggested by Ho (2003). In this case, the transition mechanism between extreme states is very simple: on each date, if the observed threshold variable of a country is less than a given value, it is called threshold parameter. Capital liquidity is defined by a specific model (or mechanism), which is different from the model used when the threshold variable is greater than the threshold parameter. For example, let’s consider a PTR model with two extreme states: the way to solve these two problems is to introduce threshold effect into the linear panel model.

Logical transfer function with a single positional parameter (M = 1)

It can be proved that the elasticity of iw.r.t s is time-varying

I think it is very intuitive for all individuals to extract these time-varying coefficients, because they show the dynamics of the relationship of interest and complement the visualization of the transfer function.

Suppose we apply this to Hansen data (4 variables instead of 2 variables, but the above formula applies). We want to study the effect of debt level on investment if we choose Tobin Q as the conversion variable. Let’s first fit the model

`PSTR(data, dep='inva', indep=4:20, indep_k=c('vala','debta','cfa','sales'),tvars=c('vala'), iT=14) `

Then calculate the time-varying coefficient and extract the Tobin Q level of the first three companies in the sample

```
for (i in 1:n){
va_i<-vala[cusip==id[i]]
g<-(1+exp(-gamma*(va_i-c)))^(-1)
tvc_i<-est[2] + mbeta*g
```

Finally, the time series are drawn

```
matplot(tvc, type = 'l', lwd=2,col = 1:3, xaxt='n'
axis(1, at=1:nrow(tvc), labels=c(1974:1987)); legend("topleft", legend =
Matplot (Vala, type ='l ', LWD = 2, col = 1:3, xaxt ='n', xlab ='year '
; axis(1, at=1:nrow(tvc), labels=c(1974:1987));
Left ("top left", left = paste ('company ', colnames (Vala), Sep ='),
```

We can see that the elasticity of investing in w.r.t debt changes over time and depends on the level of Q: the higher Q (companies with more investment opportunities), the stronger the impact. In particular, the company with the highest Q (2824) (green curve, right) shows the most stable relationship (green curve, left).

There is a problem: if the transformation variable is the same as the independent variable (or its function), the calculation of elasticity becomes more complicated. Generally, for the model with R transformation function (R + 1 mechanism), we have:

This means that the Q of w.r.t Tobin needs to be calculated in different ways.

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