Examining the Price Bubbles in Iran’s Foreign Exchange Market Using Nonlinear State Space Model: An Application of Sigma-Point Kalman Filter (SPKF)

The asset price bubble is the deviation of the asset price from its fundamental value. The price bubbles usually have common features. On one hand, providing excessive loans and facilities along with continuous raises in demand and asset price cause an inflation in price bubbles. On the other hand, the bubbles burst and the asset prices collapse because of abundant sales. Since many of the financial crisis arise from bursting bubbles of financial assets, exploring the bubbles behavior in these markets and the early detection for the prevention of adverse economic consequences is important.
Generally, the correct detection of assets price bubbles is complex and difficult due to uncertainty in accurate identification of the fundamental factors and their randomness. Especially, if the bubbles were inherent and dependent on fundamentals, the correct specification of the bubble equation will be more difficult. Many of the methods used for testing the price bubbles, have been criticized for various reasons. In this line, the present study has used Sigma-Point Kalman Filter (SPKF) to measure the price bubbles in the non-official foreign exchange market of Iran during the period of 2002:04-2015:09, forming a nonlinear state space model. In this paper, to derive the state-space model, the presented value model with variable discount rates was used as a start point. After solving this equation, the asset price includes two components, the fundamental value and the non-fundamental value or the speculative bubble. By applying the transversality condition, we obtain no bubble solution. The fundamental value is affected by market fundamentals and is expressed as the sum of the asset’s expected future dividends discounted to the present time. But, if the transversality condition be violated, this equation is the only solution for main equation.
The price bubble can be a function of fundamental factors. In addition, the relationship between the expected bubble and the current bubble values follow a nonlinear process.
If prices and fundamentals have unit root, we can write the general solution in differences. Since the bubble is an unobservable component, we regard it as a state vector and estimate it in a state space model. In addition, because fundamentals and price are observable, we consider them as observation vectors in signal equation. We assume the fundamentals follow an ARIMA (h, 1, 0) process, where h is determined empirically.
In the present study, we apply the Sigma Point Kalman filter (SPKF) for foreign exchange market in Iran. In this regard, we have taken consumer price index and real interest rate as fundamentals in nonofficial exchange rate of Iran. After specifying the state space model, this model is estimated using the maximum likelihood method. Then, we extract the bubble component using SPKF algorithm and by the MATLAB software. In order to identify the series of foreign exchange market bubbles, we have followed the SPKF methodology to detach the bubble component from the fundamental component in exchange rate. Since price variables are nonstationary in level, but stationary in first difference, we are capable of separating the bubble component of a change in the exchange rate from its fundamental components.
Regarding SPKF model estimation for the period of 2002:04 to 2015:09, most of the bubbles in exchange rate changes have been compressed at the 2012m09 so that about 61 percent of the exchange rate change can be attributed to the bubble component. Moreover, it can be seen that the non-official exchange rate of Iran in the 2000s had a very mild rising trend and slight volatilities. However, in the 2010s, at the same time with the intensification of economic sanctions, the decline in foreign exchange earnings and the rise in inflation, the exchange rate has experienced a significant growth. In this paper, we applied a Sigma-Point Kalman filter (SPKF) to extract the bubble term from exchange rate changes in non-official market of Iran during 2002:04-2015:09.The results showed that the significant part of the exchange rate changes, especially during 2010s, have been due to the formation of price bubbles. In this framework, the highest share of bubbles in exchange rate changes has been related to 2012:10 so that about 61 percent of the increase in exchange rate in this month compared to the previous month have been attributed to price bubbles. Moreover, in 2015, the exchange rate bubbles have been very insignificant and the market value of the exchange rate has been close to its fundamental value