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- *Corresponding Author:
- Lee TW

School for Engineering of Matter Transport

and Energy, Arizona State University

Tempe, AZ 85287-6106, USA

**Tel:**(480)965- 7989

**E-mail:**[email protected]

**Received Date:** June 23, 2015 **Accepted Date:** July 01, 2015 **Published Date:** July 08, 2015

**Citation: **Lee TW (2015) Non-Linear Series Inversion Method for Forecasting
Canadian GDP Growth. Bus Eco J 6:165. doi:10.4172/2151-6219.1000165

**Copyright:** © 2015 Lee TW. This is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.

**Visit for more related articles at** Business and Economics Journal

We present a new method for automatically generating mathematical models of complex, non-linear processes, and apply it for tracking and predicting the Canadian GDP. This method is derived from solving complex, non-linear problems in engineering, and is found to be an efficient method for forecasting of financial and economic variables. The method involves setting up a general non-linear series involving terms of up to 3rd-order products, where the model coefficients are systematically determined by the data on the Canadian GDP along with significant economic indicators such as currency, gross demand deposits, consumer price index and various loan rates. Results show that GDP can be predicted quantitatively and qualitatively, at various prediction intervals, with longer-term predictions showing less agreement owing to divergent dynamics in the economic variables and GDP. Other complex financial and economic processes may be analyzed and predicted using this method.

Inversion method forecasting; GDP growth

Prediction of economic parameters such as gross domestic product
(GDP) growth is important for future planning and **policy-making**.
In particular, finding the functional relationship, or a mathematical
model, between GDP growth and some key indicator variables is the
goal of many economic analyses and studies [1]. However, interaction
of the variables can be quite complex and non-linear, and some key
model parameters are not easily determined and highly variable in
time. Therefore, **several statistical **and analytical methods have been
attempted with varying degrees of success [2-8]. This is where a parallel
exists in between the current problem and many non-linear problems
in social, physical sciences and engineering: Non-linear processes are
subject to unpredictable behavior, and are difficult to model. Indeed,
methods such as neural network [1] have been applied in numerous
non-linear problems in science and **engineering**.

In this work, we describe a new method for modeling non-linear processes, called non-linear series inversion, and demonstrate its utilities. Similar to the neural network modeling, this method has been developed to predict or model the non-linear processes in dynamical systems, such as chaotic oscillator [9], and flow turbulence and chemical reactions. The method is data-driven, and automatically generates the parameters in the non-linear model. It is a generalized method, which can be applied to other complex financial and economic problems. The target parameter, to be predicted, is written as a non-linear series expansion involving the relevant variables with initially undetermined coefficients. The coefficients are then determined using a reference set of data through matrix inversion. If the correct set of variables is inserted in the non-linear series in an appropriate manner, then it is found that a reliable model can be built to predict the process variables under a wide range of initial/boundary conditions. The matrix inversion for determination of the coefficients can be done through compressive sensing algorithm, although other matrix inversion method can be employed. The significance is that this approach is applicable to a wide range of non-linear flow processes, where limited knowledge of detailed interaction between variables and unknown model coefficients can preclude or hamper accurate predictions.

Complex phenomena typically involve interactions of several or many variables. Predicting the GDP growth is also expected to involve complex interactions that cannot easily be identified or quantified. For these complex, non-linear problems, a method has been developed in this laboratory that takes into account of main key variables (such as the key economic indicators) and transforms the variables as a series of non-linear terms. Then, the coefficients of the non-linear series are found by using the actual outcome or data for the processes to be predicted.

We can start by representing the percent change of the GDP in
some time interval, as the outcome to be predicted. The factors that
will affect this percent change are many; however, we choose the key
economic indicators likely to have large impact on GDP growth/
decline. Another criterion for selecting the economic indicators is that
they should readily be available, so that implementation is straightforward
and timely. For study of GDP, the following parameters have
been widely used, and also their data are readily available at quarterly
intervals: short-term interest rate, long-term interest rate, price level
(**consumer price **index), inflation expectation, currency+gross demand
deposits. However, in many instances [1] these variables are combined
in a form to reflect trends in the economy. The combined variables are
defined as follows:

x_{1}=M_{1}/CPI

x_{2}=M_{2}/CPI

x_{3}=(R10-R90)

x_{4}=(R90-x_{p})

where M_{1}=currency+gross demand deposits

M_{2}=M_{1}+non-personal notice deposits+personal savings deposits

CPI=consumer price index

R10=average yield of Government of Canada marketable bonds, 10 years and over

R90=90-day prime corporate paper rate

x_{p}=100log_{10}(CPI/CPI_{-4})

CPI_{-4}=CPI at the previous 4^{th} quarter

These data, along with the GDP for each quarter are readily available from Statistics Canada’s CANSIM data base, and we have used the data from 1968 up to 2011. Since the magnitude of the variables vary over a large range, the data are normalized with respect to each variable value at some reference time, say 1st quarter of 1995.

The basic non-linear series that is often used is in the following form.

Here, R(t) is the percent change in Canadian GDP with respect to some reference quarter, 1, 2, 4, or 5th prior quarter. The quarter is marked with t, so that t=5 for example would be 1 year and 1 quarter from the reference quarter. The coefficients for each term in the series, is , and k, l, and m are the exponents. For some non-linear processes such as turbulence flows, this type of series worked reasonable well when the exponents were allowed to vary from 1 to 4. However, for prediction of economic processes, inclusion of exponents greater than three leads to anomalous numerical results, which indicates the degree of non-linearity (the exponents) is not very high. Thus, we only include up to cubic terms (exponent of 2) to formulate a series which consists of 32 coupled terms. If it turns out that the higher-order terms do not impact GDP, then the matrix inversion will return small coefficients for these terms. The magnitude of the coefficients in fact can indicate the dominant coupling of the parameters.

With this non-linear series and the data for Canadian GDP and other economic variables listed above, we can determine the coefficients, , through a numerical matrix inversion. For this matrix inversion, R(t) on the left-hand side of Eq.1 is the data for the percent change in Canadian GDP (from CANSIM data set), while the right-side includes the of which the products of the variables again are computed from the CANSIM data.

This way, the unknowns are the coefficients , which are obtained from matrix inversion of Equation 2. If the data are sparse, then compressive sensing algorithms are used; however, in this case there is a full set of data so that a simple matrix inversion algorithm suffices. Once the non-linear series “model” has thus been created, with the coefficients , then it can be applied for comparison with existing data and for future predictions.

First, the non-linear series coefficients are determined using the
known percent change in the Canadian GDP, from 1968 to 2010.
CANSIM publishes the quarterly Canadian GDP, from which the
percent change in one, two, or any arbitrary number of quarters, from
the current quarter, can be computed. Here, we use the entire range of
the data from 1968 to 2010 to determine the coefficients. Once these
coefficients have thus been determined, we can re-compute the percent
change in GDP using Eq 2. In **Figure 1**, we compare the computed results
(Eq 2), with the data for the percent change in two and five quarters.
It can be seen that the many of the details of the GDP percent change are reproduced by the constructed model. The accuracy is much better
for the 2-quarter predictions (predicting the GDP percent change two
quarters ahead of the given quarter, based on the economic indicator
variables at that quarter). The significant downturns in the Canadian
economy, as represented by negative percent change, in 1982 and 1990
are quite accurate tracked. Interestingly, the downturn in 2008 is not
captured, indicating that that event was unique and the correlations
between the GDP and economic indicators were unprecedented. The
upswing in the GDP is much better captured by the 2-quarter prediction
model than the 5-quarter predictions. The larger time gap, between the
current economic indicators and GDP change to be predicted after
five quarters, leads to less correlation between these parameters, and
therefore less accurate predictions. Nonetheless, the overall tracking of
the GDP change is quite good, with many of the up- and down-swings
predicted by this method.

However, even though we have used the term “predictions” to
describe the results in **Figure 1**, strictly speaking the computed results
are not true predictions, since the known GDP from 1968 to 2010 were
used to determine the coefficients in the non-linear series in Figure 1. A
true test of the methodology would be to use only the prior data, GDP
and economic indicator variables, and determine the coefficients in the
non-linear series, to calculate the GDP percent change in the quarters
ahead of time. This can be implemented by creating a moving window
of data, where we start from some initial time and use the GDP data up
to a time, say 20XX, Y^{th} quarter and use only those data to determine
the coefficients in the non-linear series, in order to predict the GDP
percent change at 20XX, (Y+1)^{th} or at some quarter ahead of 20XX, Y^{th} quarter. Then, obviously we can use the data up to any point, say 2014,
2^{nd} quarter, to predict the GDP percent change at 2014, 3^{rd} or any time
after. The results of these true predictions are plotted in **Figures 2-4 **for
+1, +2 and +4 quarter predictions, respectively.

The +1 quarter predictions shown in **Figure 2** indicate that the
current method is capable of tracking the up- and down-swings in
GDP fairly accurately. The line represents the predicted values, while
symbols are used to denote the actual data. There are some overshoots
in the predictions near 2000, 2006 and particularly 2008,
where the non-linear series model is not able to predict the sudden
downturn, due to the real-estate collapse at that time. The latter is due
to the anomalous nature of that downturn event, where economic
indicators were not correlated with the actual event. The +2 and +5
quarter predictions follow the data qualitatively, but perform less well
due to further spacing between the available data and the quarter to be
predicted. Nonetheless, it is notable that this method is able to provide
predictions that follow the actual data based on a relatively simple non- linear series inversion, without either a complex model to relate the
economic variables nor the need for any arbitrary parameters to fit the
model predictions with the data. The method is data-driven in that the
non-linear series coefficients are evaluated using a moving window of
data, and the duration of this data window is the only parameter that is
used to optimize the predictions, along with of course the form of the
non-linear series. As noted above, inclusion of higher-order terms led
to anomalous output quantities, which indicates that the interaction of
the economic variables are only weakly non-linear.

We can identify the important economic variables and their
interactions, by examining the coefficients in the non-linear series, as
shown in **Figure 5** where the first 14 coefficients are plotted. The largest
coefficients are for the i=1, 5 and 10, and these correspond to linear
terms of x_{1}=M_{1}/CPI, x_{3}=(R10-R90) and x_{4}=(R90-x_{p}), respectively. The
definition of these variables has been listed in Eq 1. The next significant
coefficients are i=2, 4, 7, 11 and 14, and these involve quadratic products
of x_{1}, x_{2}, and x_{4}. Thus, it appears that the low-order terms in the nonlinear
series dominate the expression, and higher-order terms only add
small modifications to the output results.

A method for automatically generating mathematical models of complex, non-linear processes has been applied for tracking and predicting the Canadian GDP. The method uses a non-linear series involving terms of up to 3rd-order products, where the coefficients are determined by the data on the Canadian GDP along with significant economic indicators such as currency+gross demand deposits, consumer price index and various loan rates. Results show that GDP can be predicted quantitatively and qualitatively, at various prediction intervals, with longer-term predictions showing less agreement owing to divergent dynamics in the economic variables and GDP. The significance is beyond the current applicability in predicting GDP’s. Based on the data on key variables, complex financial and economic processes may be translated into non-linear series, which adjusts its model coefficients based on the changing dynamics in the economic variables. Complex processes in finance and economy may require continuous modifications or completely new theories to be formulated. When the data involving these processes involves huge volumes and complexity, then the current method may circumvent having to invent or revise existing theories, by providing a mathematical link semi-automatically between the key variables and observable output parameters. The knowledge about the processes, limited or complete, can be input into this methodology by constructing the most plausible form of the non-linear series (such as including only the lower-order terms in this work), as opposed to having to build the mathematical model from grounds up.

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