The Dynamic Stochastic General Equilibrium (DSGE) models have become a cornerstone in understanding national economies. These models aim to encapsulate an entire economy's behavior, providing insights into how various factors interact over time. The primary objective of this article is to demonstrate the process of hand-coding a DSGE model using the R programming language, fitting it to actual data, and comparing the outcomes to those produced by established software packages such as Dynare and gEcon.

My fascination with DSGE models began last year when I encountered a job advertisement highlighting their relevance. As someone deeply interested in the applications of Bayesian statistics, I discovered that central banks often employ Bayesian methods to fit these models. This approach aligns perfectly with my interest in macroeconomic modeling, as DSGE models are specifically designed for this purpose. These models have also solidified their place in economic education, with prominent textbooks like Mankiw's Macroeconomics including sections dedicated to them.

In New Zealand, the Reserve Bank utilizes a DSGE model known as NZSim, which piques my curiosity. Furthermore, the Treasury, Te Tai hanga, employs DSGE models to forecast the economic implications of natural disasters, showcasing their practical applications.

Advocates for DSGE models assert that they are the most coherent framework for modeling macroeconomic phenomena, going so far as to dismiss those who employ alternative methodologies as dilettantes. Conversely, critics express concerns about the models' excessive mathematical complexity and what they perceive as a lack of realistic economic assumptions. This raises the critical question: who is correct? The only way to arrive at an informed conclusion is to actively engage with and analyze the following key questions:

• What exactly constitutes a DSGE model?
• What drives the development of DSGE models? What makes them more advantageous than other modeling approaches?
• How do we formulate and solve a DSGE model?
• What are the methods for fitting a DSGE model to empirical data?

While numerous software packages can fit DSGE models swiftly, truly comprehending these models necessitates hands-on computation. This article aims to meticulously walk through the stages of solving and fitting a DSGE model, comparing the results obtained through manual calculations with those generated by established software tools. To guide this exploration, I will reference the excellent PhD Macro Theory II lecture notes by Eric Sims while also articulating challenges and confusions encountered along the way, making the content accessible to those who may not be insiders in this field.

The acronym DSGE denotes Dynamic (indicating changes over time), Stochastic (involving randomness), and General Equilibrium (indicating that all markets clear). Understanding these terms is vital as we delve into the mechanics of these models.

Motivation Behind DSGE Models

To understand the motivation for developing DSGE models, let's consider a fundamental question: What are the origins of business cycles? What causes economies to oscillate between periods of growth and recession? The most straightforward illustration of business cycles is found in a toy model designed by economist Paul Samuelson in 1939, known as the Multiplier-Accelerator Interaction Model. This model is notably well-presented in Samuelson's original paper, which remains relevant even as various textbooks often overlook critical details.

In a simplified version of Samuelsons model, we can define an economy's total output, Y_t, as the sum of consumption C_t and investment I_t, leading to the equilibrium condition:

Y_t = C_t + I_t

Assuming that consumption is a fixed proportion of output, we can express this as C_t = cY_t. Additionally, we can consider a long-term steady state where Y_t = Y, I_t = I, and C_t = C, with fluctuations denoted by lower-case letters. Consequently, we can express the output as Y_t = Y + y_t.

The determination of investment is based on investors' observations of the economy's recent performance. Specifically, the difference between output in the last two periods is given by:

Y_{t-1} - Y_{t-2} = y_{t-1} - y_{t-2}

Investment deviations from the steady state can be formulated as:

i_t = a(y_{t-1} - y_{t-2})

Thus, if output has recently increased, investment tends to rise, while a decrease in output leads to a reduction in investment. This results in the equation:

y_t = 1/(1-c)(\overline{I} + (1-c)\overline{Y}) + a/(1-c)(y_{t-1} - y_{t-2})

In a more concise form, we can express this relationship as:

y_t = lpha + eta y_{t-1} - eta y_{t-2}

This model can be solved using the theory of difference equations or through graphical representation. For instance, if we set = 1 and = 0.9, starting from a boom where the last two outputs were Y₁ = Y + 1 and Y₂ = Y + 2, we can visualize the resulting business cycle graphically.

However, a significant caveat arises: if the economys participants are aware of the cyclical nature of the economy, they would adjust their investment behavior accordingly, which would theoretically eliminate the cycles. Thus, the challenge is to develop a model that allows for the recognition of business cycles while still allowing them to occurthis is precisely the problem that DSGE models aim to address.

Forward-Looking Models

In a standard DSGE model, we can express it in the form:

E_t[f(y_t, y_{t+1})] = 0

where f represents a function that includes elements of uncertainty, and y_t is a vector of economic observations. The operator E_t signifies expectations based on information available up to time t, implying that decisions made at time t depend on anticipated outcomes at time t+1.

As a preliminary exercise, we can analyze a simple model described by:

y_t = aE_t[y_{t+1}].

By iterating expectations, we derive:

y_t = a^2E_t[y_{t+2}] =  = a^k E_t[y_{t+k}].

If a < 1 and E_t[y_{t+k}] remains constrained, it leads to a condition where y_t = 0. Conversely, for a > 1, it produces a model resembling a decaying asset value, similar to a rotting fruit scenario, suggesting that y_t could potentially be non-zero.

While these models may seem overly simplistic, they serve as foundational tools for analyzing more complex models through a series of reductions.

A Model of the Economy

Lets consider a more detailed model where output, consumption, and investment at time t are related by:

Y_t = C_t + I_t.

In this case, we assume that Y_t is generated according to a Cobb-Douglas production function, where labor is assumed to be fixed:

Y_t = A_t K_t^.

Here, K_t denotes the capital stock at time t, while A_t represents the total factor productivitya measure that incorporates various factors, including technological advancements.

Capital naturally depreciates over time, represented by a rate , and is augmented through investment. This yields the law of motion for capital:

K_{t+1} = K_t - K_t + I_t.

Additionally, we assume that technology experiences random shocks, leading to fluctuations over time. These perturbations can be expressed as:

log A_{t+1} =  log A_t + _A _t

where 0 < < 1 and _t ~ N(0, 1) represents a normally distributed random variable.

The model also incorporates a representative agent whose objective is to maximize their lifetime expected utility by consuming an optimal amount at each time step. This utility function is assumed to be logarithmic, and the associated discount factor is denoted by , resulting in:

E_t[_{=t}^ ^{-t} log(C_)].

Its important to note that C_t is determined based on all available information up to time t, and the next shock _t+1 will influence C_t+1 after this knowledge is revealed. This methodology can be framed within optimal control theory as a closed-loop rather than an open-loop problem.

We can eliminate Y_t and I_t from the equations, reducing the model to:

max E_t [_{=t}^ ^{-t} log(C_)]

subject to the laws governing capital and technology:

K_{t+1} = A_t K_t^ - C_t + (1-)K_t,

log A_{t+1} =  log A_t + _A _t, _t ~ N(0, 1.

Progress on this problem can be achieved through two primary methods: solving the Bellman equation or employing Lagrange multipliers. However, incorporating expectations into the likelihood function poses a challenge. By assuming a finite number of paths for random shocks, we can simplify our expectations.

Lets say we have a numerical example with parameters set at = 0.33, = 0.99, = 0.1, = 0.95, and _A = 0.01. Following through the calculations leads us to a deep understanding of the model.

The next logical step is to explore software tools like Dynare and gEcon which facilitate the modeling process. To illustrate this, one may use the Dynare package in Matlab or Octave to fit the DSGE models, allowing for a smooth transition between theoretical calculations and practical applications.

Conclusion and Future Directions

Ultimately, fitting a DSGE model to real-world data helps economists address fundamental queries regarding the impact of policy changes, such as government spending increases. While calibration provides an intuitive method for parameter estimation, likelihood-based inference, particularly using Bayesian approaches, tends to yield more robust results.

Although DSGE models serve as powerful tools in economic modeling, they are not without their critiques. Many criticize the assumption of representative agents and the potential shortcomings in capturing the complexities and inequalities prevalent in real economies. This has led to the emergence of more nuanced models, such as the Heterogeneous Agent New Keynesian (HANK) models, which aim to provide a richer, more realistic analysis of economic behavior.

Further exploration into these advanced models, alongside the continued application of DSGE approaches, will undoubtedly enhance our understanding of economic dynamics in an increasingly complex world.