With a budget of $200 and the goal of self-hosting a 7B parameter large language model (LLM), you may need to adjust your expectations, as hosting such a model requires significant computational resources, typically beyond the capacity of a single budget GPU. However, if you're willing to compromise on performance, you could look into the following graphics cards under $200 that may suit your budget and still provide some level of support for running AI workloads:

1. NVIDIA GeForce GTX 1650 Super: This card offers good performance at a budget price point. It's built on the Turing architecture and has 4 GB of GDDR6 memory. While not explicitly designed for AI workloads, it can handle some tasks with decent performance.

2. AMD Radeon RX 5500 XT: Another budget option, the RX 5500 XT features 4 GB or 8 GB of GDDR6 memory and is built on AMD's RDNA architecture. While not as optimized for AI workloads as NVIDIA's offerings, it still delivers reasonable performance for its price.

Both cards are geared towards gaming and general use rather than large-scale AI workloads. To work with a 7B parameter LLM, you might have to rely on more powerful GPUs like NVIDIA's A100 or multiple GPUs in parallel. However, these options would far exceed your $200 budget.

You could also consider renting GPU resources from Hugging Face. This company provides resources to develop AI Models, including development tools, datasets, and GPU time (including on NVIDIA's A100 series) at more manageable price points. This could give you access to more powerful GPUs for running your LLM, but you'll need to manage your usage to stay within budget.
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